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? substitute values for a and c into the original pythagorean theorem equation. divide both sides of the equation by two, then determine the principal square root of both sides of the equation. determine the principal square root of both sides of the equation. divide both sides of the equation by 2.

Answer 1

1. We have to prove that in a 45°–45°–90° isosceled 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

$\frac{a}{\sqrt 2} = \frac{c}{2}$

$a = \frac{c}{\sqrt 2}$

So, $c = \sqrt{2}a$

6. So the hypothenuse c is $\sqrt 2$  times the length of each leg 'a'.

Answer 2

1. prove: in a 45°–45°–90° triangle, the hypotenuse is $\sqrt{2}$ times the length of each leg.
2. ecause triangle xyz is a right triangle, the side lengths must satisfy the pythagorean theorem,$a^{2} + b^{2}= c^{2}$ , which in this isosceles triangle becomes $a^{2} + a^{2}= c^{2}$
3. By combining like terms: $2a^{2}= c^{2}$
4. Determine the principal square root of both sides of the equation:
$\sqrt{2a^{2}} = \sqrt{c^{2}}$
$\sqrt{2} a=c$
5. So the hypothenuse c is $\sqrt{2}$ times the length of each leg a.