Coordinate Proofs

Grade 10 · geometry · 22 practice problems · read aloud

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What is a Coordinate Proof? 📐

A coordinate proof uses a coordinate plane and algebra to prove geometric concepts. Instead of just words and diagrams, you place shapes on a grid and use formulas (like distance, slope, and midpoint) to show that properties are true. It's useful because it gives you a precise, mathematical way to prove things about shapes.

How to Solve Problems: A Step-by-Step Guide

  1. Place the Figure: Position the shape on the coordinate plane in a way that makes calculations easy. Often, you'll place a vertex at the origin (0,0) and a side along the x-axis.
  2. Label the Coordinates: Write down the coordinates for all the key points (vertices, midpoints).
  3. State the Goal: What are you trying to prove? (e.g., "Prove two sides are equal.")
  4. Apply Formulas: Use the distance, slope, and midpoint formulas to calculate what you need.
  5. Draw a Conclusion: Compare your results to prove the statement.

Visual Examples

Example 1: Prove a Triangle is Isosceles

Given: Triangle ABC with A(0,0), B(6,0), C(3,4). Prove: It is isosceles.

  1. We need to show two sides are equal. Let's find AC and BC.
  2. Use the Distance Formula: √[(x₂-x₁)² + (y₂-y₁)²]
  3. AC = √[(3-0)² + (4-0)²] = √(9+16) = √25 = 5
  4. BC = √[(3-6)² + (4-0)²] = √(9+16) = √25 = 5
  5. Since AC = BC = 5, triangle ABC is isosceles. ✅

Example 2: Prove a Quadrilateral is a Parallelogram

Given: Quadrilateral with vertices W(0,0), X(4,0), Y(5,3), Z(1,3). Prove: It's a parallelogram.

  1. A parallelogram has opposite sides that are parallel (equal slopes).
  2. Find slopes using: m = (y₂-y₁)/(x₂-x₁)
  3. Slope WX = (0-0)/(4-0) = 0. Slope ZY = (3-3)/(5-1) = 0. So WX ∥ ZY.
  4. Slope WZ = (3-0)/(1-0) = 3. Slope XY = (3-0)/(5-4) = 3. So WZ ∥ XY.
  5. Both pairs of opposite sides are parallel, so it's a parallelogram. ✅

Common Mistakes to Avoid

Poor Placement: Placing the shape randomly, making calculations messy. Fix: Always put a vertex at (0,0) and a side on an axis.

Formula Errors: Mixing up x and y in the distance or slope formulas. Fix: Write the formula clearly and substitute carefully.

Assuming, Not Proving: Saying a shape "looks like" a parallelogram. Fix: You MUST use calculations to prove it.

Tips & Tricks

  • Right Angles: Perpendicular lines have slopes that are negative reciprocals (e.g., 2 and -1/2).
  • Midpoints: The midpoint formula averages the x's and averages the y's.
  • Strategy: Use distance formula for congruent sides. Use slope formula for parallel/perpendicular sides.

How to Practice

  • Start by proving simple things, like showing a triangle is right-angled.
  • Draw your own shapes on graph paper and prove their properties.
  • Practice using all three formulas: distance, slope, and midpoint.

Practice problems

6 of the 22, worked through step by step — try them before opening the answer.

1 Prove that quadrilateral Hana with vertices H(2,4), A(6,8), N(10,4), and A(6,0) is a square.

Hint: To prove a quadrilateral is a square, show that all four sides have equal length and that adjacent sides are perpendicular to each other. Use the distance formula for side lengths and the slope formula to check for perpendicularity.

Show the answer

Answer: The quadrilateral is a square because all four sides are equal in length (distance = sqrt(32)) and all angles are right angles (adjacent sides have slopes that are negative reciprocals).

A square is a special quadrilateral where all four sides are equal and all four angles are right angles. In coordinate geometry, we can use the distance formula to verify equal side lengths and the slope formula to verify perpendicular sides (slopes are negative reciprocals).

2 Prove that quadrilateral Hana with vertices A(2,4), B(6,8), C(10,4), and D(6,0) is a square.

Hint: To prove a quadrilateral is a square, show that all four sides are equal in length and that adjacent sides are perpendicular using the distance formula and slope calculations.

Show the answer

Answer: All four sides are equal to 4√2 and all angles are right angles, proving it is a square

A square is a special quadrilateral where all four sides are equal and all four angles are right angles. You can verify this using coordinate geometry by calculating distances between consecutive vertices and checking if the slopes of adjacent sides are negative reciprocals of each other.

3 Prove that triangle ABC with vertices A(1, 3), B(5, 7), and C(9, 3) is isosceles using coordinate geometry.

Hint: Use the distance formula to calculate the lengths of all three sides of the triangle and compare them.

Show the answer

Answer: Triangle ABC is isosceles because AB = BC = √32

An isosceles triangle has at least two sides of equal length. The distance formula can be used to determine if any two sides are equal.

4 Prove that triangle Aroha with vertices A(1, 3), R(5, 7), and H(9, 3) is isosceles using coordinate geometry.

Hint: Use the distance formula to calculate the lengths of all three sides and compare them to determine if any two sides are equal.

Show the answer

Answer: AR = RH = 4√2, so triangle Aroha is isosceles

An isosceles triangle has at least two sides of equal length. By calculating the distances between each pair of vertices using the distance formula, you can determine if this condition is met.

5 Prove that quadrilateral Sophia with vertices A(1,1), B(6,4), C(11,1), and D(6,-2) is a rhombus using coordinate geometry.

Hint: To prove a quadrilateral is a rhombus, show that all four sides have equal length and that the diagonals are perpendicular to each other.

Show the answer

Answer: All sides are equal length (5 units) and diagonals are perpendicular (slopes 0 and undefined).

A rhombus is a quadrilateral with all sides equal in length. Additionally, the diagonals of a rhombus are perpendicular bisectors of each other. You can use the distance formula to verify equal side lengths and the slope formula to check if diagonals are perpendicular.

6 Prove that quadrilateral Aroha with vertices A(9, 3), B(15, 9), C(9, 15), and D(3, 9) is a square using coordinate geometry.

  1. A) Diagonals perpendicular and one pair of equal sides
  2. B) Opposite sides parallel and one right angle
  3. C) All sides equal and diagonals equal
  4. D) All angles equal and adjacent sides equal

Hint: To prove a quadrilateral is a square, verify that all four sides have equal length and the diagonals are equal in length. Use the distance formula to calculate side lengths and diagonal lengths.

Show the answer

Answer: C) All sides equal and diagonals equal

  1. Calculate side lengths using distance formula AB = sqrt((15-9)^2 + (9-3)^2) = sqrt(6^2 + 6^2) = sqrt(36 + 36) = sqrt(72) BC = sqrt((9-15)^2 + (15-9)^2) = sqrt((-6)^2 + 6^2) = sqrt(36 + 36) = sqrt(72) CD = sqrt((3-9)^2 + (9-15)^2) = sqrt((-6)^2 + (-6)^2) = sqrt(36 + 36) = sqrt(72) DA = sqrt((9-3)^2 + (3-9)^2) = sqrt(6^2 + (-6)^2) = sqrt(36 + 36) = sqrt(72) All four sides equal sqrt(72)
  2. Calculate diagonal lengths AC = sqrt((9-9)^2 + (15-3)^2) = sqrt(0^2 + 12^2) = sqrt(144) = 12 BD = sqrt((3-15)^2 + (9-9)^2) = sqrt((-12)^2 + 0^2) = sqrt(144) = 12 Both diagonals equal 12
  3. Verify right angles using slopes Slope AB = (9-3)/(15-9) = 6/6 = 1 Slope BC = (15-9)/(9-15) = 6/(-6) = -1 Since 1 × (-1) = -1, AB ⟂ BC
  4. Conclusion All sides equal and diagonals equal, so quadrilateral Aroha is a square.
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