The hypotenuse of an isosceles right triangle
c² = 2a²: find c from a, or a from c.
Square on the hypotenuse
An isosceles right triangle has two equal sides a and a hypotenuse c. The square on the hypotenuse has double the area of the square on a leg (last lessons).
Take a = 12.
What is the area c² of the square on the hypotenuse?
What this lesson covers
The idea
In an isosceles right triangle with equal sides a and hypotenuse c, c² = 2a² (c = a√2), which gives c when a is known or a when c is known.
Square on the hypotenuse
An isosceles right triangle has two equal sides a and a hypotenuse c. The square on the hypotenuse has double the area of the square on a leg (last lessons).
Take a = 12.
What is the area c² of the square on the hypotenuse?
- 24
- 144
- 288
Find c, find a
Use the rule: the square on the hypotenuse has area 2 × a × a. Sometimes you know a and want c. Sometimes you know the area of the big square and want a.
One formula, two uses
In an isosceles right triangle with equal sides a and hypotenuse c: c² = 2a² (so c = a√2). This formula finds c when a is known, or a when c is known.
Finding c. For a = 12: c² = 2 × 12² = 288. 16² = 256 and 17² = 289, so c is between 16 and 17.
Finding a. If c = √72: 72 = 2a², so a² = 36 and a = 6.
Notes
Isosceles right triangle, equal sides a, hypotenuse c: c² = 2a². Know a? Then c² = 2 × a × a. Know c? Then a² = c² ÷ 2.
Check yourself
An isosceles right triangle has equal sides of 10 cm. What is c², the area of the square on the hypotenuse?
Answer: 200
c² = 2 × 10 × 10 = 200.
An isosceles right triangle has equal sides 7. Between which whole numbers does c lie?
The hypotenuse of an isosceles right triangle is √50. How long is each of the equal sides?
Answer: 5
50 = 2a², so a² = 25 and a = 5. Check: 2 × 5 × 5 = 50.
A square has side 9. Its diagonal is the hypotenuse of an isosceles right triangle with legs 9. The diagonal lies between two whole numbers. What is the smaller one?
Answer: 12
c² = 2 × 81 = 162. 12² = 144 and 13² = 169, so the diagonal is between 12 and 13.
- between 7 and 8. c² = 2 × 49 = 98. And 8 × 8 = 64 is below 98, so c is bigger than 8.
- between 9 and 10 — correct. Yes! c² = 98. 9² = 81 is below 98 and 10² = 100 is above it, so c is between 9 and 10 (just below 10).
- between 14 and 15. 14 = 2 × 7 is not c. c² = 98 and 14² = 196 is much too big.