To prove the length of the hypotenuse [tex]\( c \)[/tex] in a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle is [tex]\( \sqrt{2} \)[/tex] times the length of each leg, follow these steps:
1. Start with the Pythagorean Theorem:
For a right triangle, the Pythagorean theorem states that [tex]\( a^2 + b^2 = c^2 \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs of the triangle and [tex]\( c \)[/tex] is the hypotenuse.
2. Isosceles Right Triangle Properties:
In a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, the legs are of equal length. Let's denote the length of each leg by [tex]\( a \)[/tex]. Hence, the equation becomes:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
3. Combine Like Terms:
[tex]\[
2a^2 = c^2
\][/tex]
4. Solve for [tex]\( c^2 \)[/tex]:
To isolate [tex]\( a^2 \)[/tex], divide both sides of the equation by 2:
[tex]\[
a^2 = \frac{c^2}{2}
\][/tex]
5. Take the Principal Square Root:
Determine the principal square root of both sides of the equation:
[tex]\[
a = \frac{c}{\sqrt{2}}
\][/tex]
Since we want to express [tex]\( c \)[/tex] in terms of [tex]\( a \)[/tex]:
[tex]\[
c = a \times \sqrt{2}
\][/tex]
6. Conclusion:
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 summary, the length of the hypotenuse in a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle is indeed [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
