Coordinate Geometry Proofs

Grade 10 · mathematics · 100 practice problems · read aloud

🔊 Listen to this explanation

Coordinate Geometry Proofs

What Is It & Why Use It? 🤔

Coordinate proofs use a coordinate plane and algebra to prove geometric theorems. Instead of just words and diagrams, you use coordinates, slopes, and distance formulas. This method is powerful because it turns visual problems into solvable equations.

Step-by-Step Guide

  1. Place the Figure: Position the shape on the coordinate plane to make calculations easy. Use the origin (0,0), the x-axis, and y-axis.
  2. Label Coordinates: Assign coordinates to all key points (vertices, midpoints).
  3. State the Goal: Clearly write what you need to prove (e.g., "Prove two sides are congruent").
  4. Apply Formulas: Use the Distance Formula, Slope Formula, or Midpoint Formula as needed.
  5. Compare & Conclude: Compare your results and state your final conclusion.

Visual Examples

Example 1: Prove a Triangle is Isosceles

Given: Triangle with vertices A(0,0), B(4,0), C(2,3). Prove it's isosceles.

Step 1: Find side lengths using the Distance Formula.
AB = √((4-0)² + (0-0)²) = 4
AC = √((2-0)² + (3-0)²) = √(4+9) = √13
BC = √((2-4)² + (3-0)²) = √(4+9) = √13

Step 2: Compare lengths. Since AC = BC = √13, two sides are equal. Therefore, the triangle is isosceles.

Example 2: Prove a Quadrilateral is a Parallelogram

Given: Quadrilateral with vertices A(0,0), B(3,4), C(7,4), D(4,0). Prove it's a parallelogram.

Step 1: Show both pairs of opposite sides are parallel by calculating slopes.
Slope AB = (4-0)/(3-0) = 4/3
Slope DC = (0-4)/(4-7) = -4/-3 = 4/3
Slope AD = (0-0)/(4-0) = 0
Slope BC = (4-4)/(7-3) = 0

Step 2: Since AB ∥ DC and AD ∥ BC, opposite sides are parallel. Therefore, it's a parallelogram.

Common Mistakes ⚠️

Poor Placement: Placing shapes awkwardly makes math messy. Always use the origin and axes.

Formula Errors: Mixing up x and y values in formulas. Remember: Distance = √[(x₂−x₁)² + (y₂−y₁)²], Slope = (y₂−y₁)/(x₂−x₁).

Incomplete Conclusion: Don't just get an answer; state what it proves in geometric terms.

Tips & Tricks

Place Strategically: Put one vertex at (0,0) and a side along the x-axis.

Use Zeroes: Coordinates with 0 simplify calculations.

Check Parallel/Perpendicular: Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals (e.g., 2 and -1/2).

Practice Suggestions

  • Start by proving simple shapes (triangles, parallelograms).
  • Practice placing different shapes on the grid.
  • Create your own problems with easy coordinates and solve them.
  • Use graph paper to visualize the problem before calculating.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 log₂(32) + log₃(81) = ?

Hint: Remember that logarithms are exponents. For example, log₅(25) = 2 because 5² = 25.

Show the answer

Answer: 9

  1. Evaluate log₂(32) Since 2⁵ = 32, log₂(32) = 5
  2. Evaluate log₃(81) Since 3⁴ = 81, log₃(81) = 4
  3. Add the results 5 + 4 = 9

The answer is 9.

2 log₂(16) + log₃(81) = ?

Hint: Remember that logarithms are exponents. For example, log₄(64) asks '4 to what power equals 64?'

Show the answer

Answer: 8

  1. Evaluate log₂(16) - 2 to what power equals 16? 2⁴ = 16, so log₂(16) = 4
  2. Evaluate log₃(81) - 3 to what power equals 81? 3⁴ = 81, so log₃(81) = 4
  3. Add the results: 4 + 4 = 8
  4. The final answer is 8

3 log₂(64) + log₃(27) = ?

Hint: Remember that logarithms are exponents. For example, log₅(25) asks '5 to what power equals 25?'

Show the answer

Answer: 9

  1. Evaluate log₂(64) 2 to what power equals 64? 2^6 = 64, so log₂(64) = 6
  2. Evaluate log₃(27) 3 to what power equals 27? 3^3 = 27, so log₃(27) = 3
  3. Add the results 6 + 3 = 9

The answer is 9.

4 √( (5 - 1)² + (9 - 3)² ) = ?

Hint: This involves finding the distance between two points on a coordinate plane. Remember the formula for distance and simplify the expression step by step.

Show the answer

Answer: 2√13

  1. Simplify inside the parentheses** 5 - 1 = 4 9 - 3 = 6 So the expression becomes: √( (4)² + (6)² ) **
  2. Square each term** (4)² = 16 (6)² = 36 So we have: √( 16 + 36 ) **
  3. Add inside the square root** 16 + 36 = 52 So we have: √(52) **
  4. Simplify the square root** Factor 52: 52 = 4 × 13 √(52) = √(4 × 13) = √4 × √13 = 2 × √13 **
  5. Final answer** 2√13 This is the simplified form.

Let's solve step by step. We are given: √( (5 - 1)² + (9 - 3)² ) **

5 √( (5 - 1)² + (8 - 3)² ) = ?

Hint: This expression calculates the straight-line distance between two points. Remember the formula for distance in coordinate geometry and apply it carefully with the given coordinates.

Show the answer

Answer: √41

  1. Simplify inside the parentheses** First part: 5 - 1 = 4 Second part: 8 - 3 = 5 So the expression becomes: √( (4)² + (5)² ) --- **
  2. Square each term** (4)² = 16 (5)² = 25 So we have: √( 16 + 25 ) --- **
  3. Add inside the square root** 16 + 25 = 41 So we have: √(41) --- **
  4. Final answer** √41 is already simplified because 41 is a prime number. --- **Final Answer:** √41

Let's solve step-by-step. We are given: √( (5 - 1)² + (8 - 3)² ) --- **

6 √( (8 - 2)² + (13 - 5)² ) = ?

Hint: This involves finding the distance between two points using the distance formula. Remember to square the differences in coordinates before adding them.

Show the answer

Answer: 10

  1. Identify the coordinates: (x₁, y₁) = (2, 5) and (x₂, y₂) = (8, 13)
  2. Apply the distance formula: √[(x₂ - x₁)² + (y₂ - y₁)²]
  3. Calculate x₂ - x₁ = 8 - 2 = 6
  4. Calculate y₂ - y₁ = 13 - 5 = 8
  5. Square both differences: 6² = 36 and 8² = 64
  6. Add the squares: 36 + 64 = 100
  7. Take the square root: √100 = 10

The answer is 10.

Practise this topic — 10 free problems, no signup →