Certainly! Let's solve the problem step-by-step.
Step 1: Identify the type of triangle
A triangle with angles [tex]\(45^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex] is a right-angled isosceles triangle. This means it has two equal sides opposite the two [tex]\(45^\circ\)[/tex] angles.
Step 2: Represent the sides of the triangle
Let each of the equal sides (legs) of the triangle be represented by [tex]\(a\)[/tex]. The hypotenuse (the side opposite the [tex]\(90^\circ\)[/tex] angle) will be represented by [tex]\(c\)[/tex].
Step 3: Use the Pythagorean theorem
In a right-angled triangle, the Pythagorean theorem states:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
For this specific triangle, since both legs are of equal length ([tex]\(a = b\)[/tex]), the theorem simplifies to:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
Step 4: Combine like terms
Simplifying the left-hand side of the equation:
[tex]\[
2a^2 = c^2
\][/tex]
Step 5: Solve for [tex]\(c\)[/tex]
To find the length of the hypotenuse, [tex]\(c\)[/tex], solve the equation for [tex]\(c\)[/tex] by taking the square root of both sides:
[tex]\[
c = \sqrt{2a^2}
\][/tex]
[tex]\[
c = \sqrt{2} \cdot \sqrt{a^2}
\][/tex]
Since [tex]\(\sqrt{a^2} = a\)[/tex]:
[tex]\[
c = \sqrt{2} \cdot a
\][/tex]
Conclusion:
The length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]. Thus, we have shown that:
[tex]\[
c = \sqrt{2} \cdot a
\][/tex]
This verifies that the length of the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as each leg.
