Let's carefully analyze the properties of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle to determine which of the given statements is true.
First, consider the characteristic features of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle:
- This type of triangle is an isosceles right triangle, meaning that the two legs are of equal length.
- Let's denote the length of each leg of this triangle as [tex]\(L\)[/tex].
In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the hypotenuse [tex]\(H\)[/tex] relates to the legs [tex]\(L\)[/tex] by the following relationship:
[tex]\[
H = L \sqrt{2}
\][/tex]
Given this crucial observation, let's now evaluate each option:
A. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
- From the relationship [tex]\(H = L \sqrt{2}\)[/tex], we see that this statement is correct.
B. Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse.
- This would imply that [tex]\(L = H \sqrt{3}\)[/tex], which contradicts the true relationship [tex]\(H = L \sqrt{2}\)[/tex]. Therefore, this statement is incorrect.
C. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg.
- This would imply that [tex]\(H = L \sqrt{3}\)[/tex], which again contradicts the true relationship [tex]\(H = L \sqrt{2}\)[/tex]. Therefore, this statement is incorrect.
D. Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse.
- This would imply that [tex]\(L = H \sqrt{2}\)[/tex], which is the inverse of the true relationship [tex]\(H = L \sqrt{2}\)[/tex]. Therefore, this statement is incorrect.
From this analysis, the correct statement about a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is:
A. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
