If the angles of a triangle are [tex]\(45^{\circ}, 45^{\circ},\)[/tex] and [tex]\(90^{\circ}\)[/tex], show that the length of the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as each leg.

1. Substitute the side lengths of the triangle into the Pythagorean theorem.

[tex]\[a^2 + a^2 = c^2\][/tex]



Answer :

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.