To determine the true statement about a [tex]$45-45-90$[/tex] triangle, let's understand the properties of such a triangle.
A [tex]$45-45-90$[/tex] triangle is a right triangle where the two non-hypotenuse sides (legs) are equal. Let's denote the length of each leg as [tex]\( a \)[/tex].
Given the properties of a [tex]$45-45-90$[/tex] triangle:
1. The triangle has two angles of 45 degrees and one angle of 90 degrees.
2. The legs are congruent.
Using the Pythagorean theorem for the right triangle:
[tex]\[ \text{hypotenuse}^2 = \text{leg1}^2 + \text{leg2}^2 \][/tex]
Since both legs are equal:
[tex]\[ \text{hypotenuse}^2 = a^2 + a^2 \][/tex]
[tex]\[ \text{hypotenuse}^2 = 2a^2 \][/tex]
[tex]\[ \text{hypotenuse} = \sqrt{2a^2} \][/tex]
[tex]\[ \text{hypotenuse} = a\sqrt{2} \][/tex]
From this, we observe that the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
Now, let's review the given statements:
A. Each leg is [tex]\( \sqrt{3} \)[/tex] times as long as the hypotenuse.
B. Each leg is [tex]\( \sqrt{2} \)[/tex] times as long as the hypotenuse.
C. The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times as long as either leg.
D. The hypotenuse is [tex]\( \sqrt{3} \)[/tex] times as long as either leg.
The correct statement is:
C. The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times as long as either leg.
Therefore, the true statement about a [tex]$45-45-90$[/tex] triangle is:
[tex]\[ \text{Option C: The hypotenuse is } \sqrt{2} \text{ times as long as either leg.} \][/tex]
