Answer :
To determine the true statement about an isosceles right triangle, let's investigate the properties of an isosceles right triangle step-by-step.
An isosceles right triangle has two sides (legs) of the same length and one side (the hypotenuse) that is longer. The right angle is between the two legs.
1. Consider a typical isosceles right triangle where each leg has a length of [tex]\( a \)[/tex].
2. Using the Pythagorean theorem:
[tex]\[ \text{Hypotenuse}^2 = \text{Leg}^2 + \text{Leg}^2 \][/tex]
[tex]\[ \text{Hypotenuse}^2 = a^2 + a^2 \][/tex]
[tex]\[ \text{Hypotenuse}^2 = 2a^2 \][/tex]
3. Taking the square root of both sides to solve for the hypotenuse:
[tex]\[ \text{Hypotenuse} = \sqrt{2a^2} \][/tex]
[tex]\[ \text{Hypotenuse} = a\sqrt{2} \][/tex]
4. Therefore, the hypotenuse is:
[tex]\[ \sqrt{2} \text{ times as long as either leg.} \][/tex]
Given the analysis, we can see that the correct answer is:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
An isosceles right triangle has two sides (legs) of the same length and one side (the hypotenuse) that is longer. The right angle is between the two legs.
1. Consider a typical isosceles right triangle where each leg has a length of [tex]\( a \)[/tex].
2. Using the Pythagorean theorem:
[tex]\[ \text{Hypotenuse}^2 = \text{Leg}^2 + \text{Leg}^2 \][/tex]
[tex]\[ \text{Hypotenuse}^2 = a^2 + a^2 \][/tex]
[tex]\[ \text{Hypotenuse}^2 = 2a^2 \][/tex]
3. Taking the square root of both sides to solve for the hypotenuse:
[tex]\[ \text{Hypotenuse} = \sqrt{2a^2} \][/tex]
[tex]\[ \text{Hypotenuse} = a\sqrt{2} \][/tex]
4. Therefore, the hypotenuse is:
[tex]\[ \sqrt{2} \text{ times as long as either leg.} \][/tex]
Given the analysis, we can see that the correct answer is:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
