To demonstrate that in a 45-45-90 triangle (an isosceles right triangle), the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as each of the legs, we can follow these steps:
1. Understanding the triangle:
- A 45-45-90 triangle has two equal angles of [tex]\(45^\circ\)[/tex] and one right angle of [tex]\(90^\circ\)[/tex]. This means the two legs of the triangle are of equal length.
- Let's denote the length of each leg by [tex]\(a\)[/tex].
2. Substitute the side lengths into the Pythagorean theorem:
By the Pythagorean theorem, for a right-angled triangle with legs [tex]\(a\)[/tex] and [tex]\( b \)[/tex], and hypotenuse [tex]\(c\)[/tex], we have:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Since both legs are of equal length [tex]\(a\)[/tex], we can write:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
3. Combine like terms:
Combining the terms on the left side, we get:
[tex]\[
2a^2 = c^2
\][/tex]
4. Solve for [tex]\(c\)[/tex]:
To isolate [tex]\(c\)[/tex], take the square root of both sides:
[tex]\[
c = \sqrt{2a^2}
\][/tex]
Simplifying the square root gives:
[tex]\[
c = \sqrt{2} \cdot a
\][/tex]
This shows that the length of the hypotenuse [tex]\(c\)[/tex] is indeed [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex] in a 45-45-90 triangle.
