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Coordinate Geometry Proofs Test – Geometry Math (Printable PDF)

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

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Looking for a coordinate geometry proofs test? Then this Geometry test on Proving Shapes with Coordinates and Slopes checks the whole chapter, with points for each question. This test has 8 questions worth 20 points and is designed for 45 minutes. A calculator is allowed. Write exact answers unless a decimal is requested and justify every classification with computed numbers.

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

Time: 45 minutes · 8 questions · Total: 20 points · Calculator: as allowed by your teacher

Question 1 Distance and Midpoint (2 points)

Segment RS has endpoints R(−6, 2) and S(3, −4). Find its length as an exact simplified radical (1 point) and its midpoint (1 point).

Question 2 Parallel, Perpendicular or Neither (2 points)

Line a passes through (0, 4) and (6, 0). Line b passes through (−2, −5) and (4, −1). Line c passes through (1, 1) and (3, 4). Classify each pair: a and b (1 point); a and c (1 point).

Question 3 Line Through a Point (2 points)

Write, in slope-intercept form, the equation of the line through (4, −1) that is perpendicular to y = 43x + 2.

Question 4 Splitting a Segment (2 points)

Find the point P on segment AB with A(−6, 3) and B(14, −7) so that AP : PB = 2 : 3.

Question 5 Area and Perimeter (3 points)

−3−2−1123456−5−4−3−2−112xyA (−2, 1)B (5, 1)C (5, −4)
Triangle ABC.

Find the area (1 point) and the perimeter as an exact expression (2 points).

Question 6 Is It Isosceles? (3 points)

−6−5−4−3−2−11234−3−2−1123456xyK (−5, −2)L (−1, 5)M (3, −2)
Triangle KLM.

Prove that triangle KLM is isosceles and name the vertex angle.

Question 7 Prove a Parallelogram (3 points)

Prove that E(−2, 1), F(3, 0), G(6, 4) and H(1, 5) are the vertices of a parallelogram by finding the midpoints of both diagonals.

Question 8 Fourth Vertex (3 points)

Three vertices of a parallelogram are P(0, 0), Q(7, 2) and R(9, 7), taken in order. Find the fourth vertex S, and the area of the parallelogram.

View the Coordinate Geometry Proofs test answer key with scoring

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