Consider the incomplete paragraph proof.
Given: Isosceles right triangle $X Y Z\left(45^{\circ}-45^{\circ}-90^{\circ}\right.$ triangle)
Prove: In a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, the hypotenuse is $\sqrt{2}$ times the length of each leg.
Because triangle $X Y Z$ is a right triangle, the side lengths must satisfy the Pythagorean theorem, $a^2+b^2$ $=c^2$, which in this isosceles triangle becomes $a^2+a^2=c^2$. By combining like terms, $2 a^2=c^2$.
Which final step will prove that the length of the hypotenuse, $c$, is $\sqrt{2}$ times the length of each leg?
A. Substitute values for $a$ and $c$ into the original Pythagorean theorem equation.
B. Divide both sides of the equation by two, then determine the principal square root of both sides of the equation.
C. Determine the principal square root of both sides of the equation.
D. Divide both sides of the equation by 2.
Answer (1)
Start with the Pythagorean theorem for an isosceles right triangle: a 2 + a 2 = c 2 , which simplifies to 2 a 2 = c 2 .
Take the principal square root of both sides: 2 a 2 = c 2 .
Simplify the equation: 2 ⋅ a = c .
Conclude that the hypotenuse is 2 times the length of each leg: c = a 2 .
Explanation
Problem Analysis We are given an isosceles right triangle X Y Z with legs of length a and hypotenuse c . We know that a 2 + a 2 = c 2 , which simplifies to 2 a 2 = c 2 . We want to find the final step that proves c = a 2 .
Taking the Square Root To isolate c , we need to take the square root of both sides of the equation 2 a 2 = c 2 . This gives us 2 a 2 = c 2 .
Simplifying the equation Simplifying the square root, we get 2 ⋅ a 2 = c , which simplifies to 2 a = c . Therefore, c = a 2 , which means the hypotenuse is 2 times the length of each leg.
Final Answer The final step is to determine the principal square root of both sides of the equation.
Examples
Understanding the relationship between the leg and hypotenuse in a 45-45-90 triangle is useful in construction and design. For example, if you are building a square frame and need to brace it with a diagonal support, knowing that the diagonal (hypotenuse) is 2 times the side length of the square allows you to quickly calculate the required length of the support. If each side of the square is 1 meter, the diagonal support needs to be 2 ≈ 1.414 meters. This principle helps ensure structural integrity and precise measurements in various projects.
