Question

Consider the incomplete paragraph proof. Given: Isosceles right triangle XYZ (45°–45°–90° triangle) Prove: In a 45°–45°–90° triangle, the hypotenuse is times the length of each leg. Because triangle XYZ is a right triangle, the side lengths must satisfy the Pythagorean theorem, a2 + b2 = c2, which in this isosceles triangle becomes a2 + a2 = c2. By combining like terms, 2a2 = c2. Which final step will prove that the length of the hypotenuse, c, is times the length of each leg?

Answer 1

1. We have to prove that in a 45°–45°–90° isosceles triangle, the hypotenuse is times the length of each leg.

2. Since, triangle XYZ is a right triangle, the side lengths must satisfy the Pythagorean theorem, which states
`a^2+b^2=c^2`

But, in isosceles triangle it becomes
`a^2+a^2=c^2`.

3. By combining like terms, we get

`2a^2=c^2`

4. Now, we will determine the principal square root of both sides of the equation.

`\sqrt2 a = c`

5. Dividing both sides of the equation by '2', we get

`(a)/(\sqrt2)=(c)/(2)`

`a=(c)/(\sqrt 2)`

`c = \sqrt2 a`

So, the hypotenuse c is
`\sqrt 2` times the length of each leg 'a'.

Answer 2

the answer

In a 45°–45°–90° triangle, the hypotenuse is times the length of each leg. Because triangle XYZ is a right triangle, the side lengths must satisfy the Pythagorean theorem, a2 + b2 = c2, which in this isosceles triangle becomes a2 + a2 = c2. By combining like terms, 2a2 = c2.

the final step will prove that the length of the hypotenuse, c, is times the length of each leg
2a2 = c2 implies c = √2a², and since a is positive, c = a √2, because
√a² = a,
and we know that a = b, finally c = a x √2 = b x √2