Let's walk through the steps to show that the length of the hypotenuse in a right-angled isosceles triangle is [tex]$\sqrt{2}$[/tex] times the length of each leg:
Step 1: Understand the triangle type and side lengths
Given a triangle with angles [tex]$45^\circ$[/tex], [tex]$45^\circ$[/tex], and [tex]$90^\circ$[/tex], this is a right-angled isosceles triangle. In such triangles, the two leg lengths are equal. Let each leg have length [tex]$a$[/tex].
Step 2: Apply the Pythagorean theorem
In a right triangle, the Pythagorean theorem states:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Since both legs are of equal length [tex]$a$[/tex], the equation becomes:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
Step 3: Combine like terms
Combine the terms on the left-hand side:
[tex]\[
2a^2 = c^2
\][/tex]
Step 4: Solve for the hypotenuse [tex]\(c\)[/tex]
To find [tex]\(c\)[/tex], we take the square root of both sides:
[tex]\[
c = \sqrt{2a^2}
\][/tex]
[tex]\[
c = a\sqrt{2}
\][/tex]
This shows that the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the leg length [tex]\(a\)[/tex]. Therefore, the length of the hypotenuse is indeed [tex]\(\sqrt{2}\)[/tex] times the length of each leg in such a triangle.
Hence, if the leg length is [tex]\(a\)[/tex], then the hypotenuse [tex]\(c\)[/tex] evaluates numerically as [tex]\(1.4142135623730951\)[/tex] (when [tex]\(a = 1\)[/tex]), which is equivalent to [tex]\(\sqrt{2}\approx 1.4142135623730951\)[/tex].
