Let's go through this problem step-by-step to show that the length of the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as each leg in a triangle with angles [tex]\(45^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex].
1. Identify the Type of Triangle:
- The given triangle is a special type of right triangle, known as an isosceles right triangle. In an isosceles right triangle, the two legs are congruent (equal in length), and the angles are [tex]\(45^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex].
2. Let the Legs Be Represented by [tex]\(a\)[/tex]:
- Let the lengths of the two congruent legs be [tex]\(a\)[/tex].
3. Apply the Pythagorean Theorem:
- The Pythagorean theorem states that in a right triangle:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
where [tex]\(c\)[/tex] is the length of the hypotenuse, and [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the other two sides (legs).
4. Since Both Legs Are Equal:
- Here, both legs are equal, so [tex]\(b = a\)[/tex]. Plugging this into the Pythagorean theorem, we get:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
5. Combine Like Terms:
- Adding the two terms on the left side gives us:
[tex]\[
2a^2 = c^2
\][/tex]
6. Take the Principal Square Root of Both Sides:
- To find [tex]\(c\)[/tex], take the principal square root of both sides:
[tex]\[
\sqrt{2a^2} = \sqrt{c^2}
\][/tex]
7. Simplify the Expression:
- Simplifying the square roots, we get:
[tex]\[
\sqrt{2} \cdot a = c
\][/tex]
Hence:
[tex]\[
c = a\sqrt{2}
\][/tex]
Therefore, we have shown that the length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex] in a [tex]\(45^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex] triangle.
Thus, the length of the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg.
