Grade 7 Mathematics MEASUREMENT – Pythagorean Relationship Notes
MEASUREMENT — Pythagorean Relationship
Subject: Mathematics | Target age: 12 (Kenyan context)
Objective: Understand and use the Pythagorean Relationship to find the length of a missing side in a right-angled triangle.
In any right-angled triangle (one angle is 90°), the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides (called the legs).
Formula: a² + b² = c²
Where c = hypotenuse, a and b = the other two sides
- Only for right-angled triangles (one angle is 90°).
- When you know two side lengths and need to find the third.
- Make sure all lengths use the same unit (metres, centimetres).
Labels: a = vertical side, b = horizontal side, c = hypotenuse (slanted side)
Remember: c is always the longest side in a right triangle.
Example 1 (classic): Find c when a = 3 m and b = 4 m.
Step 1: Use formula a² + b² = c² → 3² + 4² = c²
Step 2: 9 + 16 = c² → 25 = c²
Step 3: c = √25 = 5 → c = 5 m
Step 2: 9 + 16 = c² → 25 = c²
Step 3: c = √25 = 5 → c = 5 m
(This is a 3–4–5 right triangle.)
Example 2: Find the missing leg if c = 13 m and one leg a = 5 m.
Use a² + b² = c² → 5² + b² = 13² → 25 + b² = 169 → b² = 169 − 25 = 144 → b = √144 = 12 m
Kenyan context problem: A ladder leans against a classroom wall in Kisumu. The ladder touches the wall at 4 m above the ground. The base of the ladder is 3 m from the wall. How long is the ladder?
We have a right angle between the ground and the wall. So a = 3 m, b = 4 m, ladder = c.
c² = 3² + 4² = 9 + 16 = 25 → c = 5 m. The ladder is 5 metres long.
- Identify the right angle and label the sides. The side opposite the right angle is c (hypotenuse).
- Write a² + b² = c².
- Substitute known values and do the arithmetic.
- If solving for a or b, rearrange: missing side² = c² − known side², then take square root.
- Check units and give final answer with units (m, cm).
- Using the formula for non-right triangles (it does not apply).
- Forgetting to take the square root at the end.
- Mixing units (metres with centimetres). Convert first.
- Find the hypotenuse if the legs are 6 m and 8 m.
- A right triangular roof has one side 7 m and the hypotenuse 25 m. Find the other side.
- A farmer has a triangular farm cornered by two straight fences meeting at right angle. One fence is 15 m and the other is 20 m. How long is the fence across (the hypotenuse)?
Answers (click to show)
1) c² = 6² + 8² = 36 + 64 = 100 → c = 10 m.
2) Missing side² = 25² − 7² = 625 − 49 = 576 → missing side = √576 = 24 m.
3) c² = 15² + 20² = 225 + 400 = 625 → c = 25 m.
2) Missing side² = 25² − 7² = 625 − 49 = 576 → missing side = √576 = 24 m.
3) c² = 15² + 20² = 225 + 400 = 625 → c = 25 m.