To prove that the length of the hypotenuse [tex]\( c \)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\( a \)[/tex] in the given isosceles right triangle, follow these steps:
1. Start with the equation derived from the Pythagorean theorem for the given isosceles right triangle:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
2. Combine like terms on the left-hand side:
[tex]\[
2a^2 = c^2
\][/tex]
3. To isolate [tex]\( a^2 \)[/tex], divide both sides of the equation by 2:
[tex]\[
a^2 = \frac{c^2}{2}
\][/tex]
4. Now, to solve for [tex]\( a \)[/tex], determine the principal square root of both sides of the equation:
[tex]\[
a = \sqrt{\frac{c^2}{2}}
\][/tex]
5. Since [tex]\(\sqrt{\frac{c^2}{2}} = \frac{c}{\sqrt{2}}\)[/tex], this step demonstrates that:
[tex]\[
a = \frac{c}{\sqrt{2}}
\][/tex]
6. By multiplying both sides by [tex]\(\sqrt{2}\)[/tex], we get:
[tex]\[
a \sqrt{2} = c
\][/tex]
Therefore, the final step that confirms that the length of the hypotenuse [tex]\( c \)[/tex] is indeed [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\( a \)[/tex] is:
Determine the principal square root of both sides of the equation.
