Which is a true statement about an isosceles right triangle?

A. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
B. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg.
C. Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse.
D. Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse.



Answer :

To solve this problem, we need to understand the properties of an isosceles right triangle. An isosceles right triangle is a right triangle in which the two legs are congruent, i.e., they have the same length.

Let's denote each leg of the triangle as [tex]\( a \)[/tex]. To find the length of the hypotenuse, we can use the Pythagorean Theorem, which states:

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

Here, [tex]\( c \)[/tex] represents the hypotenuse. Simplifying the expression on the right, we get:

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

Taking the square root of both sides to solve for [tex]\( c \)[/tex]:

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

Simplifying further:

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

Therefore, in an isosceles right triangle, the hypotenuse [tex]\( c \)[/tex] is [tex]\( \sqrt{2} \)[/tex] times the length of each leg [tex]\( a \)[/tex].

Given the options:

A. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
B. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg.
C. Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse.
D. Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse.

We determine that the correct statement based on our analysis is:

A. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.