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Difficulty: HardPythagorean Theorem and Special Right Triangles

In right triangle ABCABC, the measure of ∠B\angle B is 90∘90^\circ, the measure of ∠A\angle A is 60∘60^\circ, and the length of side ABAB is 66. Square BDEFBDEF is inscribed in △ABC\triangle ABC such that vertex DD lies on side ABAB, vertex EE lies on hypotenuse ACAC, and vertex FF lies on side BCBC. What is the length of segment BEBE?

  1. 92−369\sqrt{2} - 3\sqrt{6}Answer
  2. B
    9−339 - 3\sqrt{3}
  3. C
    36−323\sqrt{6} - 3\sqrt{2}
  4. D
    93−99\sqrt{3} - 9
  5. E
    92−339\sqrt{2} - 3\sqrt{3}

Answer

The length of segment BEBE is 92−369\sqrt{2} - 3\sqrt{6}.
The correct answer is 92−369\sqrt{2} - 3\sqrt{6}. Since △ADE\triangle ADE is a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle with short leg AD=6−xAD = 6 - x and long leg DE=xDE = x, the side length of the square is x=9−33x = 9 - 3\sqrt{3}. Segment BEBE is the diagonal of the square, which equals x2=(9−33)2=92−36x\sqrt{2} = (9 - 3\sqrt{3})\sqrt{2} = 9\sqrt{2} - 3\sqrt{6}.

Step-by-Step Solution

1
Analyze the properties of right triangle ABCABC and the inscribed square BDEFBDEF.
In △ABC\triangle ABC, ∠B=90∘\angle B = 90^\circ and ∠A=60∘\angle A = 60^\circ, so ∠C=30∘\angle C = 30^\circ. The side AB=6AB = 6. Let xx be the side length of square BDEFBDEF.
Establishing the variable xx allows us to express the dimensions of the smaller triangles formed by the square.
2
Identify the side lengths and angles of the smaller triangle △ADE\triangle ADE.
Since BD=xBD = x lies on side ABAB, AD=AB−BD=6−xAD = AB - BD = 6 - x. Because DE∥BCDE \parallel BC, ∠ADE=90∘\angle ADE = 90^\circ and ∠A=60∘\angle A = 60^\circ, making △ADE\triangle ADE a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle with DE=xDE = x.
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, the side opposite the 60∘60^\circ angle is 3\sqrt{3} times the side adjacent to the 60∘60^\circ angle.
3
Set up and solve the equation for the side length xx of the square.
Since DE=AD⋅3DE = AD \cdot \sqrt{3}, we have x=(6−x)3  ⟹  x(1+3)=63  ⟹  x=633+1=3(3−3)=9−33x = (6 - x)\sqrt{3} \implies x(1 + \sqrt{3}) = 6\sqrt{3} \implies x = \frac{6\sqrt{3}}{\sqrt{3} + 1} = 3(3 - \sqrt{3}) = 9 - 3\sqrt{3}.
Rationalizing the denominator 63(3−1)2\frac{6\sqrt{3}(\sqrt{3}-1)}{2} yields the exact side length of the square.
4
Calculate the diagonal length BEBE of square BDEFBDEF.
Segment BEBE is the diagonal of square BDEFBDEF. In a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle △BDE\triangle BDE, the hypotenuse is x2x\sqrt{2}. Thus, BE=(9−33)2=92−36BE = (9 - 3\sqrt{3})\sqrt{2} = 9\sqrt{2} - 3\sqrt{6}.
Applying the special right triangle ratio 1:1:21:1:\sqrt{2} for the square's diagonal gives the required segment length.

Key Concept

Combining 30∘−60∘−90∘30^\circ-60^\circ-90^\circ side ratios (1:3:21:\sqrt{3}:2) and 45∘−45∘−90∘45^\circ-45^\circ-90^\circ hypotenuse ratios (1:1:21:1:\sqrt{2}) to solve composite geometric figures.
Estimated Time:2m 30s
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Topics

Pythagorean Theorem and Special Right Triangles

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