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Special Right Triangles



Consider the incomplete paragraph proof.

Given: Isosceles right triangle $XYZ$ ($45^{\circ}-45^{\circ}-90^{\circ}$ 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 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, $2a^2 = c^2$.



Which final step will prove that the length of the hypotenuse is $\sqrt{2}$ times the length of each leg?


ADetermine the principal square root of both sides of the equation.
BSubstitute values for $a$ and $c$ into the original Pythagorean theorem equation.

Verified Answer :

  • Recognize that the Pythagorean theorem for an isosceles right triangle with legs \(a\) results in the equation \(2a^2 = c^2\).
  • Identify that the relationship between the hypotenuse \(c\) and leg \(a\) is found by isolating \(c\).
  • Conclude that taking the principal square root of both sides of \(2a^2 = c^2\) yields \(c = a\sqrt{2}\), which identifies Option A as the correct final step.
  • Final answer: \(\boxed{\text{A}}\)

Explanation

  1. Analyze the problem and the given equation.

An isosceles right triangle \(XYZ\) has two legs of equal length \(a\) and a hypotenuse \(c\). According to the Pythagorean theorem, which states that for any right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^2 + b^2 = c^2\). Since this triangle is isosceles, \(b = a\), so the equation becomes \(a^2 + a^2 = c^2\). Simplifying this by combining like terms gives \(2a^2 = c^2\).

  1. Determine the correct final step.

To find the relationship between the hypotenuse \(c\) and the leg \(a\), we need to solve the equation \(2a^2 = c^2\) for \(c\). By taking the square root of both sides, we get:
\[\sqrt{2a^2} = \sqrt{c^2}\]
Since \(a\) and \(c\) represent lengths, they must be positive. Thus, \(\sqrt{c^2} = c\) and \(\sqrt{2a^2} = a\sqrt{2}\). This leads to \(c = a\sqrt{2}\). Comparing this to the provided options, we see that determining the principal square root of both sides is the essential step to complete the proof.

  1. Final conclusion.

The correct final step in the proof that the hypotenuse of an isosceles right triangle is \(\sqrt{2}\) times the length of its legs is to take the principal square root of both sides of the equation \(2a^2 = c^2\). Therefore, Option A is the correct final step. The final answer is \(\boxed{\text{A}}\).

Examples

In architecture and construction, knowing that a diagonal of a square (which is composed of two \(45^{\circ}-45^{\circ}-90^{\circ}\) triangles) is \(\sqrt{2}\) times its side length is crucial. For example, if you are building a square foundation with a side length of \(10 \text{ meters}\), the diagonal distance would be \(10\sqrt{2} \approx 14.14 \text{ meters}\). This allows builders to quickly verify if the corners of a foundation are perfectly square by measuring the diagonal distance.

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