Consider the incomplete paragraph proof.
Given: Isosceles right triangle XYZ $\left(45^{\circ}-45^{\circ}-90^{\circ}\right.$ triangle)
Prove: In a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, the hypotenuse is $\sqrt{2}$ times the length of each leg.
Because triangle XYZ 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 (2)
Start with the equation 2 a 2 = c 2 .
Take the 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 XYZ (a 45-45-90 triangle) and we want to prove that the hypotenuse is 2 times the length of each leg. We are given that 2 a 2 = c 2 . The question asks for the final step to prove the relationship.
Isolating c To find the final step, we need to isolate c on one side of the equation. We start with the equation 2 a 2 = c 2 . To isolate c , we take the principal square root of both sides of the equation.
Simplifying the equation Taking the square root of both sides, we get 2 a 2 = c 2 . This simplifies to 2 ⋅ a 2 = c , which further simplifies to 2 a = c . This shows that the hypotenuse c is 2 times the length of each leg a .
Final Answer Therefore, the final step is to determine the principal square root of both sides of the equation.
Examples
In construction, if you're building a ramp that needs to have a 45-degree angle, and you know the height (one leg of the 45-45-90 triangle), you can easily calculate the length of the ramp (the hypotenuse) by multiplying the height by 2 . For example, if the ramp needs to be 3 feet high, the length of the ramp will be 3 2 feet, which is approximately 4.24 feet. This ensures the ramp has the correct slope and length.
To show that the hypotenuse c of a 4 5 ° − 4 5 ° − 9 0 ° triangle is √ 2 times the length of each leg a , we start from the equation 2 a 2 = c 2 . The final step is to take the principal square root of both sides, resulting in c = a √ 2 . Thus, the answer is option C.
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