To determine the true statement about an isosceles right triangle, let’s analyze its properties step by step.
An isosceles right triangle is a triangle with two equal sides, and the angle between these sides is [tex]\(90^\circ\)[/tex]. This creates a right triangle where the two legs (the sides of equal length) form the right angle, and the hypotenuse is the side opposite the right angle.
Given the properties of right triangles, specifically the Pythagorean Theorem (which states [tex]\(a^2 + b^2 = c^2\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the legs and [tex]\(c\)[/tex] is the hypotenuse), let's assign a value to the legs of the triangle for simplicity. Assume each leg has length [tex]\(a\)[/tex].
Because it is an isosceles right triangle, both legs are of equal length; say their length is [tex]\(a\)[/tex]. Using the Pythagorean Theorem:
[tex]\[ a^2 + a^2 = c^2 \][/tex]
[tex]\[ 2a^2 = c^2 \][/tex]
[tex]\[ c = \sqrt{2a^2} \][/tex]
[tex]\[ c = a\sqrt{2} \][/tex]
From this relationship, we can conclude that:
- The hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex].
Thus, the correct statement among the options provided is:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
Therefore, the true statement about an isosceles right triangle is indeed:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
