Geometry Coordinate Geometry and Proofs Worksheet (PDF)

Geometry math worksheet: Coordinate Geometry and Proofs – Maths PDF

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This coordinate geometry worksheet for Geometry covers coordinate geometry and proofs (Common Core G-GPE.B.4, G-GPE.B.5, G-GPE.B.6, G-GPE.B.7), with a free PDF to print.

These 20 problems practice slopes of parallel and perpendicular lines, partitioning segments, distance and coordinate proofs (Common Core G-GPE.B.4 to G-GPE.B.7). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with maps, roads and fences.

Geometry · Common Core State Standards: G-GPE.B.4, G-GPE.B.5, G-GPE.B.6, G-GPE.B.7

Coordinate geometry worksheet: exercises 1 to 10

Exercise 1 – Find a slope

Find the slope of the line through $(1, 2)$ and $(5, 10)$.

Exercise 2 – Parallel line

Parallel lines have the same slope.

Write an equation of the line through $(0, 3)$ that is parallel to $y = 2x – 1$.

Exercise 3 – Are they parallel?

Line 1 passes through $(-2, 1)$ and $(4, 4)$. Line 2 passes through $(0, 0)$ and $(6, 3)$. Are they parallel?

Exercise 4 – Perpendicular slope

A line has slope $\frac{3}{4}$. Find the slope of every line perpendicular to it.

Exercise 5 – Perpendicular line

Perpendicular slopes are opposite reciprocals.

Write an equation of the line through $(4, -1)$ that is perpendicular to $y = \frac{1}{2}x + 2$.

Exercise 6 – Are they perpendicular?

Line 1 has slope $\frac{2}{3}$ and line 2 has slope $-\frac{3}{2}$. Are the lines perpendicular?

Exercise 7 – Distance

Segment AB.

Find the distance between $A(-3, 2)$ and $B(5, 8)$.

Exercise 8 – Partition 2 : 3

P splits AB in the ratio 2 : 3 from A.

Find the point P on segment AB with $A(0, 0)$ and $B(10, 5)$ so that $AP : PB = 2 : 3$.

Exercise 9 – Partition 1 : 2

Find the point P on the segment from $A(-2, 4)$ to $B(7, -5)$ so that $AP : PB = 1 : 2$.

Exercise 10 – Partition 3 : 1

Find the point P on the segment from $A(1, 2)$ to $B(9, 10)$ so that $AP : PB = 3 : 1$.

Exercises 11 and up

Exercise 11 – Perimeter of a triangle

Triangle ABC.

Find the perimeter of triangle ABC with A(0, 0), B(4, 0) and C(4, 3).

Exercise 12 – Area of a rectangle

A rectangle has vertices $(1, 1)$, $(7, 1)$, $(7, 5)$ and $(1, 5)$. Find its area and perimeter.

Exercise 13 – Area of a triangle

Triangle ABC.

Find the area of the triangle with vertices A(0, 0), B(8, 0) and C(3, 6).

Exercise 14 – Area of a trapezoid

A trapezoid has vertices $(0, 0)$, $(6, 0)$, $(4, 4)$ and $(0, 4)$. Find its area.

Exercise 15 – Isosceles triangle

Show that the triangle with vertices $A(0, 0)$, $B(4, 0)$ and $C(2, 5)$ is isosceles.

Exercise 16 – Right angle test

Show that the triangle with vertices $A(0, 0)$, $B(6, 2)$ and $C(-1, 3)$ has a right angle at A.

Exercise 17 – Parallelogram proof

Quadrilateral ABCD.

Show that ABCD with A(0, 0), B(5, 1), C(7, 5) and D(2, 4) is a parallelogram.

Exercise 18 – Meeting point on a map

Two towns are at A(2, 3) and B(10, 11) on a map grid where one unit is 1 mile. Mei wants to meet at the point that is $\frac{3}{4}$ of the way from A to B. Find the meeting point and its distance from A to the nearest tenth of a mile.

Exercise 19 – New road

An existing road runs through points (0, 4) and (5, 2). A new road must be perpendicular to it. What slope should the new road have?

Exercise 20 – Fence cost

A triangular lot has vertices (0, 0), (12, 0) and (12, 9) with coordinates in feet. Fencing costs $3.50 per foot. Find the perimeter and the cost of fencing the whole lot.

Key facts behind this coordinate geometry worksheet

Key facts

Parallel lines: equal slopes. Perpendicular lines: slopes multiply to -1.
Point P with AP:PB = a:b is a/(a+b) of the way from A to B.
Distance: square root of (x2 – x1) squared plus (y2 – y1) squared.
Perimeter = sum of distances; area uses base times height or decomposition.
Coordinate proofs test slopes (parallel, perpendicular) and distances (equal sides).

How to use this coordinate geometry worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the coordinate geometry worksheet as a PDF with the button below and keep it with your notes.

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