To prove that in a \(45^\circ-45^\circ-90^\circ\) triangle, the hypotenuse is \(\sqrt{2}\) times the length of each leg, we start by considering the fundamental properties and applying the correct mathematical steps.
Given:
- We have an isosceles right triangle \(XYZ\) with angles \(45^\circ, 45^\circ, 90^\circ\).
- Let the length of each leg of this triangle be \(a\).
Step-by-step solution:
1. Apply the Pythagorean Theorem:
Since triangle \(XYZ\) is a right triangle, the side lengths must satisfy the Pythagorean theorem:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Here, both legs \(a\) and \(b\) are of equal length. Therefore, we can write:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
2. Combine like terms:
By combining like terms, we get:
[tex]\[
2a^2 = c^2
\][/tex]
3. Solve for \(c^2\) in terms of \(a^2\):
To isolate \(a^2\), divide both sides of the equation by \(2\):
[tex]\[
a^2 = \frac{c^2}{2}
\][/tex]
4. Determine the principal square root of both sides:
By taking the square root of both sides, we get:
[tex]\[
a = \frac{c}{\sqrt{2}}
\][/tex]
5. Solve for the hypotenuse \(c\):
Rearrange the equation to solve for \(c\):
[tex]\[
c = a \sqrt{2}
\][/tex]
Therefore, the length of the hypotenuse \(c\) in a \(45^\circ-45^\circ-90^\circ\) triangle is \(\sqrt{2}\) times the length of each leg \(a\). This final step confirms that the hypotenuse \(c\) is indeed \(\sqrt{2}\) times the length of each leg.
Thus, the statement "Divide both sides of the equation by two, then determine the principal square root of both sides of the equation." completes the paragraph proof.
