Coordinate Geometry Proofs Test Answer Key and Scoring

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

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Looking for a coordinate geometry proofs test answer key? Then this answer key gives the solutions and the scoring guide of the Geometry test on Proving Shapes with Coordinates and Slopes. Each solution shows the formula, the substitution and the conclusion. The scoring notes show how points are earned for the setup, the arithmetic and the proof statement.

Standards: G-GPE.B.4, G-GPE.B.5, G-GPE.B.6, G-GPE.B.7 · Level: Grade 10 Geometry

Sommaire

Question 1 Distance and Midpoint (2 points)

RS = √(9² + (−6)²) = √117 = 3√13 ≈ 10.82.

Midpoint = (−6 + 32, 2 + −42) = (−1.5, −1).

3√13; (−1.5, −1)

Scoring: 1 point each.

Question 2 Parallel, Perpendicular or Neither (2 points)

Slope a = 0 − 46 − 0 = −23. Slope b = −1 − (−5)4 − (−2) = 46 = 23. Slope c = 32.

a and b: −23 · 23 = −49 — neither parallel nor perpendicular.

a and c: −23 · 32 = −1 — perpendicular.

a, b: neither; a, c: perpendicular

Scoring: 1 point per pair.

Question 3 Line Through a Point (2 points)

The perpendicular slope is −34. Then −1 = −34(4) + b = −3 + b, so b = 2.

y = −34x + 2

Scoring: 1 point for the slope, 1 point for the intercept.

Question 4 Splitting a Segment (2 points)

P is 25 of the way from A to B. Changes: 20 in x, −10 in y.

x: −6 + 25(20) = 2. y: 3 + 25(−10) = −1.

P = (2, −1)

Scoring: 1 point for the setup with the fraction 2/5, 1 point for the coordinates.

Question 5 Area and Perimeter (3 points)

Legs 7 and 5.

AB = 7 and BC = 5 with a right angle at B. Area = ½ · 7 · 5 = 17.5. AC = √74.

Area = 17.5; perimeter = 12 + √74

Scoring: 1 point for area, 1 point for √74, 1 point for the perimeter.

Question 6 Is It Isosceles? (3 points)

KL = √(4² + 7²) = √65. LM = √(4² + (−7)²) = √65. KM = √(8² + 0²) = 8.

KL = LM, so the triangle is isosceles with vertex angle at L.

Isosceles; vertex angle at L

Scoring: 1 point for each of the two equal sides, 1 point for the vertex.

Question 7 Prove a Parallelogram (3 points)

Diagonal EG midpoint = (−2 + 62, 1 + 42) = (2, 2.5). Diagonal FH midpoint = (3 + 12, 0 + 52) = (2, 2.5).

The diagonals share a midpoint, so EFGH is a parallelogram.

Scoring: 1 point for each midpoint, 1 point for the conclusion.

Question 8 Fourth Vertex (3 points)

S = P + R − Q = (0 + 9 − 7, 0 + 7 − 2) = (2, 5). Check: the diagonals PR and QS both have midpoint (4.5, 3.5).

Enclose the parallelogram in the rectangle from (0, 0) to (9, 7), area 63. The leftover corners are triangles of area 7, 5, 7 and 5, plus two 2-by-2 squares of area 4 each, for a total of 32. So the area is 63 − 32 = 31.

S = (2, 5); area = 31

Scoring: 1 point for S, 1 point for the check, 1 point for the area.

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