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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.



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, $2a^2=c^2$.



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


ASubstitute values for $a$ and $c$ into the original Pythagorean theorem equation.
BDivide both sides of the equation by two, then determine the principal square root of both sides of the equation.
CDetermine the principal square root of both sides of the equation.
DDivide both sides of the equation by 2.

Verified Answer :

  • Identify the Pythagorean theorem for an isosceles right triangle as \(2a^2 = c^2\).
  • Recognize that the principal square root is the inverse operation for isolating \(c\).
  • Apply the square root to both sides to derive \(c = a\sqrt{2}\).
  • Conclude that the final step is to determine the principal square root of both sides of the equation.
  • The final answer is \(\boxed{\text{Determine the principal square root of both sides of the equation}}\).

Explanation

  1. Analyze the problem and given equation.

The problem focuses on the relationship between the legs and the hypotenuse of an isosceles right triangle (45°-45°-90°). Let the legs of the triangle be denoted by \(a\) and the hypotenuse by \(c\). According to the Pythagorean theorem, which states \(a^2 + b^2 = c^2\), we substitute the length of the legs \(a = b\), resulting in \(a^2 + a^2 = c^2\). Combining the terms gives the equation \(2a^2 = c^2\). Our goal is to isolate \(c\) to prove the relationship \(c = a\sqrt{2}\).

  1. Determine the necessary algebraic operation.

To isolate \(c\) in the equation \(2a^2 = c^2\), we need to perform an operation that reverses the squaring of \(c\). The inverse of squaring is taking the square root. By applying the principal square root to both sides of the equation, we maintain the equality because lengths \(a\) and \(c\) are positive. This operation yields \(\sqrt{2a^2} = \sqrt{c^2}\). Applying the property of radicals \(\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}\), the left side simplifies to \(\sqrt{2} \cdot \sqrt{a^2} = a\sqrt{2}\), and the right side simplifies to \(c\). This directly results in the equation \(c = a\sqrt{2}\).

  1. Conclude with the correct step.

Comparing the algebraic steps with the provided options, we see that determining the principal square root of both sides of the equation \(2a^2 = c^2\) is the most direct and efficient method to solve for \(c\). Thus, the correct step to prove that the hypotenuse is \(\sqrt{2}\) times the leg length is to determine the principal square root of both sides of the equation. The final result is that the length of the hypotenuse is \(c = a\sqrt{2}\).

Examples

Consider a carpenter building a square table. To ensure the corners are perfectly square, they measure the diagonal of the table. If each side of the table (the legs of the resulting right triangle) is 1 meter, they know the diagonal (the hypotenuse) should be \(\sqrt{2} \times 1 \approx 1.414\) meters. By applying the relationship \(c = a\sqrt{2}\), they can quickly check the accuracy of their construction without needing complex tools.

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