Full solution

Q. An isosceles right triangle has a hypotenuse of length

58

inches. What is the perimeter, in inches, of this triangle?

\newline

(A)

29\sqrt{2}

\newline

(B)

58\sqrt{2}

Understand Triangle Properties: Understand the properties of an isosceles right triangle. An isosceles right triangle has two equal sides, which are the legs, and one hypotenuse. The legs meet at a right angle. Since the triangle is isosceles, the lengths of the legs are equal. To find the length of the legs, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse $c$ is equal to the sum of the squares of the lengths of the other two sides $a$ and $b$ . $c^2 = a^2 + b^2$ Since $a = b$ in an isosceles right triangle, we can write: $c^2 = 2a^2$
Calculate Leg Length: Calculate the length of one leg of the triangle. $\newline$ We know the hypotenuse $c$ is $58$ inches, so we can substitute this value into the equation from Step $1$ to find the length of one leg $a$ . $\newline$ $58^2 = 2a^2$ $\newline$ $3364 = 2a^2$ $\newline$ Divide both sides by $2$ to solve for $a^2$ : $\newline$ $a^2 = \frac{3364}{2}$ $\newline$ $a^2 = 1682$ $\newline$ Now, take the square root of both sides to find $a$ : $\newline$ $58$ $0$
Correct Calculation Mistake: Realize there is a mistake in the calculation and correct it. $\newline$ The correct calculation should be: $\newline$ $a^2 = \frac{3364}{2}$ $\newline$ $a^2 = 1682$ $\newline$ However, $1682$ is not a perfect square, and we should simplify the square root to get the exact length of the leg. $\newline$ $a = \sqrt{1682}$ is incorrect. $\newline$ We need to find the square root of $1682$ in a simplified form.