Let's analyze each option given about an isosceles right triangle.
First, recall some key properties of an isosceles right triangle:
- An isosceles right triangle has two equal legs.
- The angles opposite those legs are each 45 degrees.
- The hypotenuse is opposite the right angle (90 degrees) and is the longest side of the triangle.
Consider an isosceles right triangle where both legs are of length [tex]\( a \)[/tex]. We can use the Pythagorean theorem to find the hypotenuse.
The Pythagorean theorem states that in a right triangle:
[tex]\[ a^2 + a^2 = h^2 \][/tex]
where [tex]\( h \)[/tex] is the hypotenuse.
Simplifying,
[tex]\[ 2a^2 = h^2 \][/tex]
[tex]\[ h = \sqrt{2a^2} \][/tex]
[tex]\[ h = a\sqrt{2} \][/tex]
This indicates that the hypotenuse [tex]\( h \)[/tex] is [tex]\(\sqrt{2} \)[/tex] times the length of either leg.
Now, let's evaluate the options:
A. Each leg is [tex]\( \sqrt{3} \)[/tex] times as long as the hypotenuse.
- This does not align with our result, so this statement is false.
B. The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times as long as either leg.
- This matches the result, where [tex]\( h = a\sqrt{2} \)[/tex]. This statement is true.
C. Each leg is [tex]\( \sqrt{2} \)[/tex] times as long as the hypotenuse.
- This is the inverse of our result and does not hold true. This statement is false.
D. The hypotenuse is [tex]\( \sqrt{3} \)[/tex] times as long as either leg.
- This does not align with our result, so this statement is false.
Therefore, the correct and true statement about an isosceles right triangle is:
B. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
