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.
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.
