Geometry  ·  MGU Mathematics Series
Part V: Coordinate Geometry, Transformations, and Modeling
Distance · Midpoint · Slope · Coordinate Proof · Classification
Chapter 13 of 14
MGU Mathematics Series  ·  Geometry  ·  Library Textbook Edition

Chapter 13:
Coordinate Geometry
and Algebraic Proof

The distance formula, midpoint formula, slope, equations of lines, classification of triangles and quadrilaterals, and coordinate proof — algebra as the language of geometry

■ Part V of 5  ·  Chapter 13 of 14 ■

■ 14 Sections  •  13 Learning Objectives ■

Follows Chapters 1–12  ·  Prepares for Chapter 14: Transformations, Symmetry, and Geometric Modeling

Chapter 13 — Front Matter

Purpose and Overview

Geometry began in this book with points, lines, planes, angles, and diagrams drawn on paper. Since then, you have classified triangles, proven congruence and similarity, worked with right triangles and trigonometry, studied polygons and circles, and calculated area and volume. Each of those topics used geometric reasoning — reasoning based on shapes, relationships, and logical argument.

Chapter 13 brings a powerful new tool to all of that work: the coordinate plane. When you place a geometric figure on a coordinate plane, every point becomes an ordered pair, every segment becomes a measurable distance, every midpoint becomes a calculable location, and every relationship between lines becomes a relationship between slopes. Suddenly, algebra can do what diagrams could only suggest: it can measure, verify, and prove geometric facts with arithmetic precision.

This chapter is the clearest bridge in the entire book between Algebra I and Geometry. You will review the distance formula, midpoint formula, and slope, then use them as tools for geometric reasoning. You will classify triangles and quadrilaterals using coordinates. You will write coordinate proofs — formal arguments based on algebraic calculations rather than on diagram reasoning alone. And you will prove congruence and similarity using the algebra of coordinates.

Opening Question

A city planner is designing a park. She places the park's boundary on a coordinate grid, with corners at A(0, 0), B(6, 0), C(6, 4), and D(0, 4). She wants to know: What is the length of each side? Where is the center of the park? Is the park a rectangle, or only a parallelogram? Are the diagonals equal in length?

Every one of these questions can be answered using only algebra — the distance formula, the midpoint formula, and slope. No protractor, no compass. Just coordinates and calculation. This is what coordinate geometry does. It turns spatial questions into algebraic ones.

14Sections
13Objectives
14Key Terms
12+Worked Examples

Why This Chapter Matters. Coordinate geometry is the foundation of analytic geometry, calculus, physics, engineering, architecture, computer graphics, robotics, surveying, GPS, navigation, and data science. You have already used coordinates in Algebra I. In this chapter, you learn to use them as a geometric tool — for classifying shapes, measuring relationships, and writing formal proof.

Learning Objectives

  1. 1Explain what coordinate geometry is and why it connects algebra to geometry.
  2. 2Use the distance formula to find lengths of segments on the coordinate plane.
  3. 3Use the midpoint formula to find the midpoint of a segment on the coordinate plane.
  4. 4Calculate slope and interpret it in a geometric context.
  5. 5Identify parallel and perpendicular lines using slope.
  6. 6Write equations of lines and use them in geometric reasoning.
  7. 7Classify triangles on the coordinate plane by side length and slope.
  8. 8Classify quadrilaterals on the coordinate plane using distance, slope, and midpoint.
  9. 9Write coordinate proofs for geometric theorems.
  10. 10Use coordinates to prove triangle congruence and similarity.
  11. 11Apply coordinate geometry to real-world measurement and modeling problems.
  12. 12Identify and avoid common coordinate geometry mistakes.
  13. 13Describe how coordinate geometry prepares for analytic geometry, precalculus, and calculus.
Chapter 13 — Contents

Sections at a Glance

Sections 13.1 – 13.4
The Three Coordinate Tools: Distance, Midpoint, and Slope
Why Coordinates Matter · Distance Formula · Midpoint Formula · Slope
Section 13.1

Why Coordinate Geometry Matters

Pure geometry is powerful: it describes shape, proves relationships, and reasons from diagrams. But it works in the abstract. When you say that triangle ABC is congruent to triangle DEF, you have proven a logical relationship — but you cannot measure the exact position of each vertex, the exact length of each side, or the exact coordinates of the centroid without a measurement system.

The coordinate plane supplies that measurement system. When a geometric figure is placed on a coordinate plane, every point gets an address — an ordered pair (x, y). Once points have addresses, distances become computable, midpoints become findable, slopes become calculable, and equations become available to describe every line, circle, and curve. This is the central idea: coordinates turn geometric relationships into algebraic relationships.

Geometric QuestionAlgebraic Answer
How long is this segment?Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²]
Where is the halfway point?Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Are these lines parallel?Compare slopes: m₁ = m₂
Do these sides meet at 90°?Check: m₁ · m₂ = −1
What shape is this quadrilateral?Use distance, slope, and midpoint together
Historical Note

The coordinate system we use was formalized by the French mathematician René Descartes in the seventeenth century. Before Descartes, algebra and geometry were studied as entirely separate fields. His insight — that curves can be described by equations and equations can be visualized as curves — made calculus possible. The system is named the Cartesian coordinate system in his honor.

Section 13.2

Distance Formula Review

The distance formula is simply the Pythagorean Theorem applied to the coordinate plane. If you want to find the distance between two points P₁(x₁, y₁) and P₂(x₂, y₂), build a right triangle: draw a horizontal leg of length |x₂ − x₁| and a vertical leg of length |y₂ − y₁|. The segment P₁P₂ is the hypotenuse.

The Distance Formula
d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]
Derived from a² + b² = c². Squaring removes all negative signs, so order of subtraction does not affect the result.
Diagram Instruction

Plot P₁(1, 2) and P₂(5, 5) on a coordinate plane. Draw a horizontal dashed segment from P₁ to (5, 2), and a vertical dashed segment from (5, 2) to P₂. Label the horizontal leg 4 and the vertical leg 3. Label the hypotenuse d. Mark the right angle at (5, 2). This is the right triangle whose hypotenuse gives the distance between P₁ and P₂.

Example 13.2.1
Finding a segment length — whole number answer
  • ProblemFind the length of AB if A(2, 1) and B(8, 9).
  • ThinkI know two endpoints. I will apply the distance formula directly.
  • Solve d = √[(8 − 2)² + (9 − 1)²] = √[6² + 8²] = √[36 + 64] = √100 = 10
  • Check6-8-10 is a Pythagorean triple (a multiple of 3-4-5). The answer is reasonable.
Answer: AB = 10
Example 13.2.2
Finding an exact length in radical form
  • ProblemFind the distance between P(−3, 2) and Q(1, −4).
  • ThinkThe answer may not simplify to a whole number. I will leave it in simplified radical form.
  • Solve d = √[(1 − (−3))² + (−4 − 2)²] = √[4² + (−6)²] = √[16 + 36] = √52 = √(4 · 13) = 2√13
  • Check√52 ≈ 7.2. The horizontal distance is 4 and the vertical distance is 6, so a distance of about 7.2 is reasonable.
Answer: PQ = 2√13
Student Warning

Do not add the differences before squaring. The formula requires squaring each difference separately: (x₂ − x₁)² + (y₂ − y₁)² ≠ (x₂ − x₁ + y₂ − y₁)². Adding first is one of the most common distance formula errors.

Practice 13.2Distance Formula
  1. A(0, 0) and B(6, 8)
  2. P(2, 3) and Q(7, 15)
  3. C(−1, 4) and D(5, −2)
  4. M(0, −3) and N(−4, 0)
  5. X(−2, −5) and Y(4, 3)
  6. G(3, 3) and H(3, −7)
Section 13.3

Midpoint Formula Review

The midpoint of a segment is the point exactly halfway between the two endpoints. On a number line, the midpoint between two values is their average. The same idea extends to two dimensions: take the average of the x-coordinates and the average of the y-coordinates.

The Midpoint Formula
M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )
Average the x-coordinates; average the y-coordinates. The result is the coordinates of the midpoint.
Example 13.3.1
Finding a midpoint
  • ProblemFind the midpoint of CD if C(4, 6) and D(10, 2).
  • Solve M = ((4 + 10)/2, (6 + 2)/2) = (14/2, 8/2) = (7, 4)
  • CheckIs (7, 4) between C(4, 6) and D(10, 2)? The x-coordinate 7 is between 4 and 10. The y-coordinate 4 is between 6 and 2. Yes. ✓
Answer: The midpoint of CD is (7, 4).
Example 13.3.2
Using the midpoint to find a missing endpoint
  • ProblemThe midpoint of EF is M(3, −1). One endpoint is E(1, 2). Find F.
  • ThinkI know M and E. Set up two equations using the midpoint formula — one for x, one for y.
  • Solve x: (1 + x₂)/2 = 3 → 1 + x₂ = 6 → x₂ = 5 y: (2 + y₂)/2 = −1 → 2 + y₂ = −2 → y₂ = −4
  • CheckMidpoint of (1, 2) and (5, −4): M = ((1+5)/2, (2+(−4))/2) = (3, −1) ✓
Answer: F = (5, −4)
Tip — Finding a Missing Endpoint

To find a missing endpoint, write the midpoint formula as two separate equations — one for x and one for y — and solve each individually. This keeps the algebra clean and reduces errors.

Practice 13.3Midpoint Formula
  1. A(4, 6) and B(10, 2)
  2. P(−3, 1) and Q(7, 5)
  3. C(0, 0) and D(5, 9)
  4. E(−6, −2) and F(2, 8)
  5. Midpoint (4, 3), one endpoint (1, −1). Find the other.
  6. Midpoint (0, 2), one endpoint (3, 5). Find the other.
Section 13.4

Slope in Geometry

Slope measures how steep a line is and in what direction it rises or falls. In Algebra I, you used slope to describe linear equations. In geometry, slope is a tool for describing relationships between lines — whether they are parallel, perpendicular, or neither.

Slope Formula
m = (y₂ − y₁) / (x₂ − x₁)
Rise divided by run. The order of subtraction must be the same in both numerator and denominator.
Slope ValueGeometric Meaning
Positive slopeLine rises from left to right
Negative slopeLine falls from left to right
Zero slopeLine is horizontal
Undefined slopeLine is vertical (x₂ = x₁, division by zero)
Example 13.4.1
Calculating slope
  • ProblemFind the slope of the line through A(2, 5) and B(6, 11).
  • Solvem = (11 − 5) / (6 − 2) = 6/4 = 3/2
Answer: The slope of line AB is 3/2.
Student Warning

The order of subtraction must be consistent. If you write y₂ − y₁ in the numerator, you must write x₂ − x₁ in the denominator — using the same labeling for both. Writing m = (y₂ − y₁)/(x₁ − x₂) produces the wrong sign.

Practice 13.4Slope
  1. (1, 3) and (4, 9)
  2. (−2, 5) and (3, −5)
  3. (0, 0) and (6, 4)
  4. (2, 7) and (2, −3)
  5. (−1, 4) and (5, 4)
  6. (3, −1) and (7, 5)
Sections 13.5 – 13.6
Parallel Lines, Perpendicular Lines, and Equations
Equal Slopes · Negative Reciprocals · Slope-Intercept Form · Point-Slope Form
Section 13.5

Parallel and Perpendicular Lines

Parallel Lines Theorem

Two nonvertical lines are parallel if and only if they have the same slope.

ℓ₁ ∥ ℓ₂ ⟺ m₁ = m₂

Two vertical lines are parallel to each other even though their slopes are undefined.

Perpendicular Lines Theorem

Two nonvertical lines are perpendicular if and only if the product of their slopes is −1.

ℓ₁ ⊥ ℓ₂ ⟺ m₁ · m₂ = −1

Equivalently: each slope is the negative reciprocal of the other. If m₁ = 2/3, then m₂ = −3/2.

Example 13.5.1
Identifying parallel and perpendicular lines
  • ProblemThree lines pass through these pairs of points. Identify any parallel or perpendicular pairs.
    Line 1: (0, 1) and (3, 5)  ·  Line 2: (1, 3) and (4, 7)  ·  Line 3: (−2, 4) and (2, 1)
  • Solve m₁ = (5−1)/(3−0) = 4/3 m₂ = (7−3)/(4−1) = 4/3 m₃ = (1−4)/(2−(−2)) = −3/4
  • CompareLine 1 and Line 2: m₁ = m₂ = 4/3 → Lines 1 and 2 are parallel.
    Lines 1 & 3: m₁ · m₃ = (4/3)(−3/4) = −1 → Lines 1 and 3 are perpendicular (Line 2 ⊥ Line 3 also).
Answer: Lines 1 ∥ Line 2. Lines 1 and 2 ⊥ Line 3.
Tip — Finding the Negative Reciprocal

To find the negative reciprocal of a fraction, flip it and change its sign. For example: 5/2 becomes −2/5. The number −3 (written as −3/1) becomes 1/3. Always check: the product of the original slope and its negative reciprocal must equal −1.

Section 13.6

Equations of Lines in Geometry

FormEquationBest When
Slope-intercepty = mx + bYou know the slope and y-intercept
Point-slopey − y₁ = m(x − x₁)You know the slope and one point
Example 13.6.1
Writing an equation through two points
  • ProblemWrite the equation of the line through A(2, 3) and B(6, 7).
  • ThinkFind slope first, then use point-slope form.
  • Solve m = (7 − 3)/(6 − 2) = 4/4 = 1 Using point A(2, 3): y − 3 = 1(x − 2) y = x + 1
  • CheckDoes B(6, 7) satisfy y = x + 1? → 7 = 6 + 1 = 7 ✓
Answer: y = x + 1
Example 13.6.2
Writing a perpendicular line through a point
  • ProblemLine ℓ has equation y = 2x − 3. Write the equation of the line through P(4, 1) that is perpendicular to ℓ.
  • ThinkSlope of ℓ is 2. Perpendicular slope is −1/2.
  • Solve y − 1 = −(1/2)(x − 4) y − 1 = −x/2 + 2 y = −x/2 + 3
  • CheckProduct of slopes: 2 · (−1/2) = −1 ✓. At P(4, 1): y = −(4/2) + 3 = −2 + 3 = 1 ✓
Answer: y = −(1/2)x + 3
Practice 13.5 – 13.6Parallel, Perpendicular, and Line Equations
  1. Line 1 through (0, 1) and (2, 5); Line 2 through (1, 0) and (3, 4). Parallel, perpendicular, or neither?
  2. Line 1 through (0, 3) and (4, 1); Line 2 through (2, 0) and (4, 4). Parallel, perpendicular, or neither?
  3. Write the equation of the line with slope 3 through (2, 1).
  4. Write the equation of the line through (0, 4) and (3, −2).
  5. Write the equation perpendicular to y = 4x − 1 through (8, 2).
  6. Write the equation parallel to y = −2x + 3 through (1, 5).
Sections 13.7 – 13.8
Classifying Triangles and Quadrilaterals
Scalene · Isosceles · Equilateral · Right Triangles · Parallelograms · Rectangles · Rhombi
Section 13.7

Classifying Triangles on the Coordinate Plane

From Chapter 4, you know triangles can be classified by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse). On the coordinate plane, you can classify triangles using only the distance formula and slope. Use the distance formula to compare side lengths. Use perpendicular slopes to identify right angles.

Example 13.7.1
Classifying by side length
  • ProblemClassify △PQR with P(0, 0), Q(4, 0), R(2, 3) by its sides.
  • ThinkFind PQ, QR, and PR using the distance formula, then compare.
  • Solve PQ = √[(4−0)² + (0−0)²] = √16 = 4 QR = √[(2−4)² + (3−0)²] = √[4 + 9] = √13 PR = √[(2−0)² + (3−0)²] = √[4 + 9] = √13
  • CompareQR = PR = √13 and PQ = 4 ≠ √13. Exactly two sides are equal.
Answer: △PQR is isosceles because QR = PR.
Example 13.7.2
Classifying as a right triangle using slope
  • ProblemDetermine whether △ABC with A(1, 1), B(5, 1), C(5, 4) is a right triangle.
  • ThinkFind slopes of all three sides and check for perpendicularity.
  • Solve m_AB = (1−1)/(5−1) = 0 (horizontal) m_BC = (4−1)/(5−5) = 3/0 (undefined — vertical) m_AC = (4−1)/(5−1) = 3/4
  • ObserveAB is horizontal; BC is vertical. A horizontal line ⊥ a vertical line, so AB ⊥ BC.
Answer: △ABC is a right triangle with the right angle at B.
Tip — Double-Check with the Pythagorean Theorem

After computing side lengths for classification, you can verify a right triangle without slopes by checking whether the three lengths satisfy a² + b² = c². If the two shorter sides squared add to the longest side squared, the triangle is a right triangle.

Section 13.8

Classifying Quadrilaterals on the Coordinate Plane

Property to VerifyCoordinate Tool
Opposite sides are parallelEqual slopes
Opposite sides are equalEqual distances
All sides are equalAll four distances equal
Diagonals bisect each otherMidpoints of both diagonals are the same
Diagonals are equalEqual diagonal lengths
Diagonals are perpendicularSlopes of diagonals multiply to −1
Right angles existConsecutive sides have perpendicular slopes
Example 13.8.1
Classifying a quadrilateral
  • ProblemClassify quadrilateral ABCD with A(0, 0), B(4, 0), C(5, 3), D(1, 3).
  • ThinkCompute slopes of all four sides. Check for parallel pairs.
  • Solve m_AB = 0, m_DC = 0 → AB ∥ DC m_AD = 3, m_BC = 3 → AD ∥ BC
  • Check rectanglm_AB · m_AD = 0 · 3 = 0 ≠ −1. No right angles.
  • Check rhombusAB = 4, AD = √(1 + 9) = √10. Opposite sides equal but AB ≠ AD.
Answer: ABCD is a parallelogram — both pairs of opposite sides parallel and equal, but not a rectangle or rhombus.
Example 13.8.2
Verifying a rectangle
  • ProblemVerify that EFGH with E(0, 0), F(5, 0), G(5, 3), H(0, 3) is a rectangle.
  • Solve m_EF = 0 (horizontal), m_GH = 0 (horizontal) → EF ∥ GH m_FG = undefined (vertical), m_EH = undefined (vertical) → FG ∥ EH Horizontal ⊥ vertical → all four angles are right angles. Diagonals: EG = √(25 + 9) = √34 FH = √(25 + 9) = √34 → Diagonals are equal.
Answer: EFGH is a rectangle. Both pairs of opposite sides parallel, all right angles, and equal diagonals.
Practice 13.7 – 13.8Classification
  1. Classify △ABC with A(0,0), B(5,0), C(2,4) as scalene, isosceles, or equilateral.
  2. Determine whether △PQR with P(1,1), Q(4,1), R(4,5) is a right triangle. Identify the right angle vertex.
  3. Given PQRS with P(0,0), Q(6,0), R(6,4), S(0,4): classify as parallelogram, rectangle, rhombus, or square. Justify with calculations.
  4. Given WXYZ with W(0,0), X(3,4), Y(8,4), Z(5,0): classify the quadrilateral. Show all needed slopes and distances.
Sections 13.9 – 13.11
Coordinate Proof, Congruence, and Similarity
Setting Up Proofs · Using Variables · SSS Congruence · Scale Factor · SSS Similarity
Section 13.9

Coordinate Proofs

A coordinate proof is a geometric proof that uses the coordinate plane. Instead of working with abstract diagrams and listing theorems, a coordinate proof assigns coordinates to key points and uses algebra to verify a geometric relationship. Coordinate proofs must apply to all cases, not just one example — so variables are used for coordinates whenever the figure's dimensions are not fixed.

Setup Strategies for Coordinate Proofs

Place one vertex at the origin — simplifies expressions because one or both coordinates are zero.

Place one side along the x-axis — makes slope of that side zero, simplifying comparisons.

Use variables for unknown lengths — the proof holds for all valid values, not just one case.

Use even numbers for coordinates — midpoint formula divides by 2, so even coordinates avoid fractions.

Proof 13.9.1
The Triangle Midsegment Theorem — coordinate proof
  • Given△ABC with midpoints M on AB and N on AC.
  • ProveMN ∥ BC and MN = (1/2)BC.
  • SetupLet A = (0, 0), B = (2b, 0), C = (2c, 2d). Using 2b, 2c, 2d ensures midpoints are clean.
  • Midpoints M = midpoint of AB = ((0 + 2b)/2, (0 + 0)/2) = (b, 0) N = midpoint of AC = ((0 + 2c)/2, (0 + 2d)/2) = (c, d)
  • Slopes m_MN = (d − 0)/(c − b) = d/(c − b) m_BC = (2d − 0)/(2c − 2b) = 2d/[2(c − b)] = d/(c − b) → m_MN = m_BC, so MN ∥ BC ✓
  • Lengths MN = √[(c − b)² + d²] BC = √[(2c − 2b)² + (2d)²] = √[4(c − b)² + 4d²] = 2√[(c − b)² + d²] → MN = (1/2)BC ✓
Reasoning Notes: This proof works for every triangle — not just the one sketched — because b, c, and d are arbitrary real numbers. The use of 2b, 2c, 2d prevented fractions from cluttering the midpoint calculation. The proof is now complete. □
Proof 13.9.2
The diagonals of a rectangle are equal in length
  • SetupLet the rectangle have vertices A(0, 0), B(a, 0), C(a, b), D(0, b) where a > 0 and b > 0.
  • Diagonals AC = √[(a − 0)² + (b − 0)²] = √[a² + b²] BD = √[(0 − a)² + (b − 0)²] = √[a² + b²]
  • ConcludeAC = BD = √(a² + b²). The diagonals of the rectangle are equal in length. □
Reasoning Notes: The key is placing the rectangle with sides along the axes. This gives all four vertices simple coordinates. The proof holds for any rectangle, since a and b can be any positive values.
Section 13.10

Proving Congruence with Coordinates

Two triangles are congruent if all three pairs of corresponding sides are equal in length. This is the SSS Congruence criterion applied to coordinates. Two segments are congruent if their distances are equal.

Example 13.10.1
Proving triangles congruent by SSS
  • ProblemShow that △PQR with P(0,0), Q(4,0), R(2,3) is congruent to △XYZ with X(1,1), Y(5,1), Z(3,4).
  • Solve △PQR PQ = √[16 + 0] = 4 QR = √[(2−4)² + 9] = √13 PR = √[4 + 9] = √13
  • Solve △XYZ XY = √[(5−1)² + 0] = 4 YZ = √[(3−5)² + (4−1)²] = √13 XZ = √[(3−1)² + (4−1)²] = √13
  • ComparePQ = XY = 4, QR = YZ = √13, PR = XZ = √13. All three pairs of corresponding sides are equal.
Answer: △PQR ≅ △XYZ by SSS.
Proof 13.10.2
The diagonals of a parallelogram bisect each other
  • SetupLet parallelogram have vertices A(0, 0), B(a, 0), C(a + c, d), D(c, d). This makes AB ∥ DC and AD ∥ BC.
  • Midpoints Midpoint of diagonal AC: ((0 + a + c)/2, (0 + d)/2) = ((a+c)/2, d/2) Midpoint of diagonal BD: ((a + c)/2, (0 + d)/2) = ((a+c)/2, d/2)
  • ConcludeBoth diagonals share the midpoint ((a+c)/2, d/2). Therefore the diagonals of the parallelogram bisect each other. □
Section 13.11

Proving Similarity with Coordinates

Two triangles are similar if all three ratios of corresponding side lengths are equal. On the coordinate plane, compute all six side lengths, then verify that the ratios form a constant scale factor k.

Example 13.11.1
Proving similarity using side length ratios
  • ProblemShow △ABC with A(0,0), B(4,0), C(0,3) is similar to △DEF with D(0,0), E(8,0), F(0,6).
  • Sides △ABCAB = 4, BC = √(16+9) = 5, CA = 3
  • Sides △DEFDE = 8, EF = √(64+36) = 10, FD = 6
  • Ratios DE/AB = 8/4 = 2 EF/BC = 10/5 = 2 FD/CA = 6/3 = 2
Answer: △ABC ∼ △DEF with scale factor k = 2 by SSS Similarity.
MGU Library Connection
  • Similarity through coordinates is the foundation for dilations — studied next in Chapter 14.
  • Trigonometric ratios depend on consistent ratios of sides in similar right triangles: see Chapter 8 of this textbook and the MGU Trigonometry and Precalculus course.
  • Coordinate similarity extends to higher-dimensional vector spaces in MGU Linear Algebra.
  • Conic sections (circles, ellipses, parabolas, hyperbolas) are defined as coordinate equations in MGU Algebra II and Precalculus.
Practice 13.9 – 13.11Coordinate Proof Practice
  1. Proof 1. Let △ABC have A(0, 0), B(2a, 0), C(0, 2b). M = midpoint of AB, N = midpoint of AC. Prove MN ∥ BC and MN = (1/2)BC.
  2. Proof 2. Use the general parallelogram A(0, 0), B(a, 0), C(a+c, d), D(c, d). Prove both diagonals share the same midpoint.
  3. Proof 3. Place a rhombus at A(0, b), B(a, 0), C(0, −b), D(−a, 0). Prove the diagonals are perpendicular.
  4. Proof 4. △ABC with A(0, 0), B(6, 0), C(3, 5). Use the distance formula to prove △ABC is isosceles.
  5. Show that △PQR with P(0,0), Q(5,0), R(0,12) is similar to △XYZ with X(0,0), Y(10,0), Z(0,24). State the scale factor.
Sections 13.12 – 13.14
Applications, Common Mistakes, and the Road Ahead
Real-World Coordinate Geometry · Eight Common Errors · Analytic Geometry and Precalculus Preview
Section 13.12

Coordinate Geometry Applications

Application 13.12.1
Finding the center of a field
  • ProblemA soccer field has corners at A(0, 0), B(100, 0), C(100, 60), D(0, 60) in meters. Find the center.
  • SolveCenter = midpoint of diagonal AC = ((0+100)/2, (0+60)/2) = (50, 30)
Answer: The center of the field is at (50, 30).
Application 13.12.2
Verifying a right angle in construction
  • ProblemA carpenter has two beams. Beam 1 runs from (0, 0) to (6, 2). Beam 2 runs from (6, 2) to (8, −4). Are the beams perpendicular at their junction?
  • Solve m₁ = 2/6 = 1/3 m₂ = (−4−2)/(8−6) = −6/2 = −3 m₁ · m₂ = (1/3)(−3) = −1 ✓
Answer: Yes. The beams are perpendicular at their junction.
Application 13.12.3
Navigation distance
  • ProblemA ship is at A(2, 5) and travels to port B(14, 10), where each unit is 1 nautical mile. How far does the ship travel?
  • SolveAB = √[(14−2)² + (10−5)²] = √[144 + 25] = √169 = 13
Answer: The ship travels 13 nautical miles.
Practice 13.12Real-World Applications
  1. Architecture: A building foundation has corners at A(0, 0), B(30, 0), C(30, 20), D(0, 20) in feet. Find the length of each side, the length of each diagonal, and the center of the foundation.
  2. Hiking: A hiker starts at (0, 0) and reaches a waterfall at (5, 12) in kilometers. How far did the hiker travel in a straight line? What is the midpoint of the path?
  3. Fire Stations: Three fire stations are at F₁(1, 2), F₂(7, 2), and F₃(4, 7). A hospital is at H(4, 4). Which station is closest to the hospital?
  4. Robotics: A robot arm reaches from A(0, 0) to B(9, 40). How long must the arm be? Where is the midpoint of the arm's range?
  5. Surveying: A property has three corners at P₁(0, 0), P₂(12, 0), and P₃(6, 8) where each unit = 10 meters. Find the perimeter of the triangular property in meters.
Section 13.13

Common Coordinate Proof Mistakes

Mistake 1 — Using specific numbers in a general proof
Proving a theorem only for one specific example (e.g., a rectangle with a = 5, b = 3).

Why it is wrong: A proof must apply to all cases. One example illustrates but does not prove. Use variables a, b, c, d so the proof holds for all valid values.

Mistake 2 — Adding differences before squaring
d = √[(x₂ − x₁) + (y₂ − y₁)]² instead of √[(x₂−x₁)² + (y₂−y₁)²]

Square each difference separately. Adding them first produces a completely different calculation.

Mistake 3 — Confusing midpoint with distance
Using the distance formula when asked for a midpoint location (or vice versa).

Midpoint gives a location — an ordered pair. Distance gives a length — a single number. Read the problem carefully before choosing a formula.

Mistake 4 — Checking only one pair of sides for parallelogram
Showing one pair of opposite sides is parallel and concluding the figure is a parallelogram.

A trapezoid also has one pair of parallel sides. Show both pairs of opposite sides are parallel to conclude parallelogram.

Mistake 5 — Computing slope inconsistently
m = (y₂ − y₁) / (x₁ − x₂) — denominator order is reversed.

The order of subtraction must be the same in both numerator and denominator. Switching the denominator order produces the wrong sign for the slope.

Mistake 6 — Building the conclusion into the setup
Placing a "general rectangle" with all four corners at obviously right-angle positions, then claiming the rectangle property is proven.

A coordinate proof should use coordinates that represent a general figure of the given type, not one with the desired properties already forced in. Use offset vertices (e.g., (a + c, d)) for parallelograms.

Mistake 7 — Concluding parallel from a diagram alone
Looking at the diagram and writing "these segments appear parallel" without computing slopes.

In a coordinate proof, always verify parallel relationships algebraically. Appearances can deceive, especially with oblique figures.

Mistake 8 — Leaving the proof conclusion unstated
Computing all relevant distances or slopes but ending the proof without stating what was proven.

Every coordinate proof should end with a sentence stating the geometric conclusion: "Therefore, MN ∥ BC and MN = (1/2)BC." The square symbol □ marks the end.

Practice 13.13Error Analysis
  1. A student finds the distance between (2, 3) and (6, 7) as d = √[(6+2)² + (7+3)²] = √164. What is the error? What is the correct answer?
  2. A student classifies quadrilateral ABCD by computing only the slope of AB and DC, finding both zero, and concluding it is a parallelogram. What is wrong with this reasoning?
  3. To prove all rectangles have equal diagonals, a student uses the specific rectangle with vertices (0,0), (5,0), (5,3), (0,3) and writes "Proof complete." What is missing?
  4. A student computes the slope through (1, 4) and (3, 10) as m = (3−1)/(10−4) = 1/3. What is the error? What is the correct slope?
  5. A student concludes △ABC ∼ △DEF because AB/DE = 2 and BC/EF = 2 but did not compute CA/FD. Is the conclusion valid? Why or why not?
Section 13.14

Preparing for Analytic Geometry and Precalculus

Chapter 13 has established the algebraic language of geometry: distance, midpoints, slope, equations of lines, and coordinate proof. You are now ready to use those tools in the most dynamic setting the book offers.

Chapter 14 extends coordinate geometry into the study of transformations — motions that move, reflect, rotate, and scale geometric figures. Every transformation rule is a coordinate rule:

TransformationCoordinate Rule
Translation(x, y) → (x + a, y + b)
Reflection over x-axis(x, y) → (x, −y)
Rotation 90° counterclockwise(x, y) → (−y, x)
Dilation by factor k(x, y) → (kx, ky)

Beyond this book, coordinate geometry expands in two directions. Toward more geometry: analytic geometry studies conic sections — circles, ellipses, parabolas, and hyperbolas — as equations on the coordinate plane, in Algebra II and Precalculus. Toward higher dimensions: linear algebra uses vectors, matrices, and coordinate systems in three or more dimensions. Machine learning uses distances and slopes in spaces with hundreds of dimensions.

The central idea to carry forward: algebra and geometry are the same subject viewed through two different lenses. Coordinates make the connection between them precise, verifiable, and usable for every kind of mathematical and scientific work.

Practice 13.14Readiness Check
  1. A segment has endpoints A(1, 3) and B(7, 11). (a) Find AB. (b) Find the midpoint of AB. (c) Write the equation of line AB. (d) Write the equation of the perpendicular bisector of AB.
  2. Classify △RST with R(0,0), S(6,0), T(3, 3√3) as scalene, isosceles, or equilateral. Show all work.
  3. The transformation rule (x, y) → (x + 3, y − 2) moves all points 3 units right and 2 units down. Apply this rule to the triangle with vertices A(1, 4), B(5, 4), C(3, 7). What are the new coordinates? Has the shape or size changed?
Chapter 13 — Back Matter

Chapter Summary

Key Terms Review

coordinate plane
A flat surface defined by a horizontal x-axis and a vertical y-axis intersecting at the origin. Every point has an address (x, y).
ordered pair
A pair of numbers (x, y) that identifies the location of a point on the coordinate plane.
distance formula
d = √[(x₂−x₁)² + (y₂−y₁)²]. Gives the length of the segment between two points. Derived from the Pythagorean Theorem.
midpoint formula
M = ((x₁+x₂)/2, (y₁+y₂)/2). Gives the point exactly halfway between two endpoints by averaging each coordinate.
slope
m = (y₂−y₁)/(x₂−x₁). The ratio of rise to run. Measures the steepness and direction of a line.
parallel lines
Lines in the same plane that never intersect. Verified on the coordinate plane by equal slopes: m₁ = m₂.
perpendicular lines
Lines that intersect at a right angle. Verified on the coordinate plane by the condition m₁ · m₂ = −1.
negative reciprocal
The slope of a line perpendicular to a given line. To find it, flip the given slope and change its sign.
slope-intercept form
y = mx + b. The equation of a line written with slope m and y-intercept b.
point-slope form
y − y₁ = m(x − x₁). The equation of a line written using a known point and the slope.
coordinate proof
A proof that uses algebraic calculations on coordinates to establish a geometric relationship. Variables are used so the proof applies to all cases.
scale factor
The constant ratio k by which all corresponding sides of two similar figures are related. Found by dividing corresponding side lengths.
collinear
Three or more points that lie on the same line. Verified by showing they satisfy the same linear equation or have the same slope between every pair.
SSS Similarity
Two triangles are similar if all three ratios of corresponding side lengths are equal (the same constant k for all three pairs).

Concept Review Questions

  1. What is the relationship between the distance formula and the Pythagorean Theorem? Sketch the right triangle that explains the connection.
  2. Explain why the midpoint formula averages the coordinates of the two endpoints.
  3. What does a slope of −3 tell you about how a line behaves on the coordinate plane?
  4. How do you determine whether two lines are parallel using only their slopes?
  5. How do you determine whether two lines are perpendicular using their slopes?
  6. Why is it important to use variables — not specific numbers — when setting up a coordinate proof for a general theorem?
  7. What three formulas are most commonly used in coordinate proofs, and what does each calculate?
  8. Explain how you would classify a triangle as isosceles using coordinates.
  9. Explain how you would verify that a quadrilateral is a rhombus using coordinates.
  10. What is the difference between proving congruence and proving similarity on the coordinate plane?

Skill Practice

Part ADistance and Midpoint
  1. A(0,0) and B(6,8)
  2. P(2,3) and Q(7,15)
  3. C(−1,4) and D(5,−2)
  4. Midpoint of E(4,6) and F(10,2)
  5. Midpoint of P(−3,1) and Q(7,5)
  6. Midpoint (4,3), endpoint (1,−1). Find the other endpoint.
Part BSlope and Line Equations
  1. Slope through (1,3) and (4,9)
  2. Slope through (−2,5) and (3,−5)
  3. Slope through (2,7) and (2,−3)
  4. Equation through (2,1) with slope 3
  5. Equation through (0,4) and (3,−2)
  6. Perpendicular to y=4x−1 through (8,2)
Part CClassification
  1. Classify △RST with R(0,0), S(5,0), T(0,5) by sides and angles.
  2. Classify △ABC with A(0,0), B(4,0), C(2, 2√3) by sides. (All sides equal?)
  3. Classify ABCD with A(0,0), B(5,0), C(5,4), D(0,4). Show all work.
  4. Classify WXYZ with W(0,0), X(4,3), Y(7,3), Z(3,0). Compute all needed slopes and distances.
Part DCoordinate Proof
  1. Prove that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. Use A(0,0), B(2a,0), C(0,2b).
  2. Prove that a triangle with vertices A(0,0), B(6,0), C(3,5) is isosceles.
  3. Prove that △ABC with A(0,0), B(4,0), C(0,3) is similar to △DEF with D(0,0), E(8,0), F(0,6). State the scale factor.

Word Problems

  1. Architecture: A designer places the corners of a triangular atrium at A(0,0), B(12,0), and C(6,9) on a coordinate grid (units in feet). Find the perimeter. Is the atrium triangle equilateral, isosceles, or scalene?
  2. Engineering: Two support cables attach to the same anchor on a tower at (0,10). Cable 1 goes to the ground at (6,0). Cable 2 goes to the ground at (−6,0). Are the cables equal in length? Verify algebraically.
  3. Surveying: A surveyor places boundary markers at P₁(0,0), P₂(8,0), and P₃(4,6), each unit representing 50 meters. Find the perimeter in meters. Classify the triangle.
  4. City Planning: A city planner designs a rectangular park with corners at A(0,0), B(200,0), C(200,80), D(0,80) in meters. Find the length of each diagonal and the coordinates of the center of the park.
  5. Navigation: A plane flies from airport A(0,0) to airport B(90,120) (units in miles). A refueling stop is placed at the midpoint of the route. What are the coordinates of the refueling stop? How far is the refueling stop from A?
  6. Design: A graphic designer places two lines on a poster. Line 1 passes through (0,5) and (4,9). Line 2 passes through (1,0) and (5,4). Are they parallel, perpendicular, or neither? Write both equations.

Chapter 13 Checkpoint

Chapter 13 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 13
Part 1 — Formulas
  1. Write the distance formula and explain what each variable represents.
  2. Write the midpoint formula and explain what each variable represents.
  3. Write the slope formula and explain what "rise over run" means in a geometric context.
Part 2 — Computation
  1. Find the distance between A(−3, 1) and B(5, 7).
  2. Find the midpoint of CD with C(2, −4) and D(8, 6).
  3. Find the slope of the line through (0, 5) and (4, −3).
  4. Are lines with slopes 3/4 and −4/3 parallel, perpendicular, or neither?
  5. Write the equation of the line through (1, 2) with slope 5.
Part 3 — Classification
  1. Classify △ABC with A(0,0), B(5,0), C(2,4) as scalene, isosceles, or equilateral. Show all work.
  2. Given PQRS with P(0,0), Q(6,0), R(6,4), S(0,4): determine whether PQRS is a parallelogram, rectangle, rhombus, or square. Justify with calculations.
Part 4 — Coordinate Proof
  1. Prove that the segment connecting the midpoints of two sides of a right triangle is parallel to the third side. Use A(0,0), B(2a,0), C(0,2b).
  2. Given that the diagonals of quadrilateral WXYZ share the same midpoint, what can you conclude about WXYZ? Explain.
Part 5 — Application
  1. A surveyor places three property corners at P₁(0,0), P₂(12,0), P₃(6,8) on a coordinate grid where each unit represents 10 meters. Find the perimeter of the triangular property in meters.
  2. A bridge deck runs from A(0,20) to B(80,20). A diagonal cable runs from C(0,0) to D(80,40). Are the deck and cable parallel, perpendicular, or neither?
Part 6 — Error Analysis
  1. A student computes the midpoint of (3,5) and (7,9) as (10,14). Identify and correct the error.
  2. A student proves a specific rectangle has equal diagonals and writes "Proof complete." What is still missing for a complete coordinate proof of the general theorem?
Looking Ahead
Chapter 14: Transformations, Symmetry, and Geometric Modeling

Chapter 13 gave you the algebraic language of geometry. Chapter 14 will use it to study transformations — rules that move, reflect, rotate, and scale figures on the coordinate plane.

A translation shifts all coordinates by a fixed amount: (x, y) → (x + a, y + b). A reflection flips figures across a line of symmetry. A rotation turns figures around a center point. A dilation scales figures by a factor k. Every one of these operations is a coordinate rule you can apply, compose, and reason about algebraically.

Chapter 14 will also connect geometry to art, design, robotics, computer graphics, physics, and engineering — showing that geometry is not merely a class you take, but a system of ideas that describes the structure of the physical and digital world.

MGU Library Connection
  • Coordinate Geometry Formula Sheet — Appendix M of this textbook
  • Transformation Rules Reference — Appendix N of this textbook (covered in Chapter 14)
  • Algebra I, Chapter 5: Graphing Linear Equations (slope, intercepts, coordinate plane fundamentals)
  • Algebra II: Conic Sections — circles, ellipses, parabolas, and hyperbolas as coordinate equations
  • Trigonometry and Precalculus: Unit circle, polar coordinates, parametric equations, vectors
  • MGU Calculus I: Limits, derivatives as slopes, definite integrals as areas — all on the coordinate plane
  • MGU Physics: Position, velocity, force, and field representations as coordinate vectors
  • MGU Linear Algebra: Coordinate geometry extended to n-dimensional vector spaces