Pythagoras Theorem
ICSE Class 7 · Mathematics · Chapter 16
What this chapter covers
- The right-angled triangle and its sides
- Squares of numbers: a quick refresher
- The Pythagorean relation
- Verifying the theorem by measuring
- Verifying the theorem with squares and areas
- Pythagorean triplets
- Finding the hypotenuse
- Finding a leg
- Problems on Pythagoras theorem
Key points
- A right-angled triangle has one angle of exactly 90°.
- The hypotenuse is the side opposite the right angle and is always the longest side; the other two sides are the legs.
- Pythagoras theorem: in a right-angled triangle, (hypotenuse)2 = (leg)2 + (leg)2, or c2 = a2 + b2.
- The theorem can be verified by drawing and measuring, by counting squares on squared paper, by cutting and fitting paper squares, and with a knotted rope.
- In area form, the square on the hypotenuse equals the two squares on the legs put together.
- A Pythagorean triplet is three natural numbers with a2 + b2 = c2, such as 3, 4, 5 and 5, 12, 13; any multiple of a triplet is also a triplet.
- The numbers 2m, m2 - 1 and m2 + 1 form a triplet for any natural number m greater than 1.
- To find the hypotenuse, add the squares of the legs; to find a leg, subtract the square of the known leg from the square of the hypotenuse.
- Three lengths make a right-angled triangle only when the square of the longest equals the sum of the squares of the other two.
Questions answered
In △ XYZ, ∠ X = 90°. Name the hypotenuse and the legs.
The hypotenuse is YZ, the side opposite X. The legs are XY and XZ.
Why is the hypotenuse always the longest side of a right-angled triangle?
The longest side of a triangle faces its largest angle. In a right-angled triangle the largest angle is the right angle, and the hypotenuse faces it.
State Pythagoras theorem in words.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The legs of a right-angled triangle are 15 cm and 20 cm. Find the hypotenuse.
152 + 202 = 225 + 400 = 625, and 25 × 25 = 625, so the hypotenuse is 25 cm.
The hypotenuse of a right-angled triangle is 41 cm and one leg is 9 cm. Find the other leg.
412 - 92 = 1681 - 81 = 1600, and 40 × 40 = 1600, so the other leg is 40 cm.
Which of these are Pythagorean triplets: (a) 6, 8, 10 (b) 7, 8, 9 (c) 12, 16, 20?
(a) 36 + 64 = 100, yes. (b) 49 + 64 = 113, not 81, no. (c) 144 + 256 = 400, yes.
Use the rule 2m, m2 - 1, m2 + 1 with m = 7 to write a Pythagorean triplet.
14, 48, 50. Check: 196 + 2304 = 2500 = 502.
A ladder 10 m long reaches a window 8 m above the ground. How far is the foot of the ladder from the wall?
102 - 82 = 100 - 64 = 36, so the distance is 6 m.