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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?

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Nirina7
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

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 [tex]a^2+b^2=c^2[/tex]

But, in isosceles triangle it becomes [tex]a^2+a^2=c^2[/tex].

3. By combining like terms, we get

[tex]2a^2=c^2[/tex]

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

[tex]\sqrt2 a = c[/tex]

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

[tex]\frac{a}{\sqrt2}=\frac{c}{2}[/tex]

[tex]a=\frac{c}{\sqrt 2}[/tex]

[tex]c = \sqrt2 a[/tex]

So, the hypotenuse c is [tex]\sqrt 2[/tex]  times the length of each leg 'a'.

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