Certainly! Let's go through the steps to show that the length of the hypotenuse in a 45°-45°-90° triangle is [tex]\(\sqrt{2}\)[/tex] times as long as each leg.
### Step-by-Step Solution
1. Identify the Triangle Configuration:
A triangle with angles 45°, 45°, and 90° is an isosceles right triangle. This means it has two sides of equal length (the legs) and a hypotenuse.
2. Use the Pythagorean Theorem:
For a right triangle, the relationship between the legs and the hypotenuse is given by the Pythagorean theorem:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Since the two legs are of equal length, let's denote them both as [tex]\(a\)[/tex]. Hence, the equation becomes:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
3. Combine Like Terms:
Combine the [tex]\(a^2\)[/tex] terms on the left side of the equation:
[tex]\[
2a^2 = c^2
\][/tex]
4. Solve for [tex]\(c\)[/tex]:
To find the length of the hypotenuse [tex]\(c\)[/tex], take the square root of both sides of the equation:
[tex]\[
\sqrt{2a^2} = \sqrt{c^2}
\][/tex]
Simplifying this, we get:
[tex]\[
\sqrt{2} \cdot a = c
\][/tex]
Thus:
[tex]\[
c = \sqrt{2} \cdot a
\][/tex]
This shows that the length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex].
### Conclusion
Thus, in a 45°-45°-90° triangle, the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg.
