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Difficulty: MediumPythagorean Theorem and Special Right Triangles

In the figure below, quadrilateral ABCDABCD is composed of two adjacent right triangles, △ABD\triangle ABD and △BCD\triangle BCD, sharing side BDBD. In △ABD\triangle ABD, the right angle is at vertex AA, and the measure of ∠ABD\angle ABD is 60∘60^\circ. In △BCD\triangle BCD, the right angle is at vertex CC, and the measure of ∠BDC\angle BDC is 45∘45^\circ. If the length of segment ABAB is 66 inches, what is the length, in inches, of segment BCBC?

  1. 626\sqrt{2}Answer
  2. B
    636\sqrt{3}
  3. C
    1212
  4. D
    12212\sqrt{2}
  5. E
    434\sqrt{3}

Answer

The length of segment BCBC is 626\sqrt{2} inches.
In the 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle ABDABD, short leg AB=6AB = 6 gives hypotenuse BD=12BD = 12. In the 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle BCDBCD, hypotenuse BD=12BD = 12 gives leg BC=122=62BC = \frac{12}{\sqrt{2}} = 6\sqrt{2} inches.

Step-by-Step Solution

1
Analyze △ABD\triangle ABD using 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle ratios.
Since ∠A=90∘\angle A = 90^\circ and ∠ABD=60∘\angle ABD = 60^\circ, ∠ADB=30∘\angle ADB = 30^\circ. The side opposite 30∘30^\circ is AB=6AB = 6. Therefore, the hypotenuse BD=2×AB=2(6)=12BD = 2 \times AB = 2(6) = 12.
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, the hypotenuse is twice the shorter leg.
2
Analyze △BCD\triangle BCD using 45∘−45∘−90∘45^\circ-45^\circ-90^\circ special right triangle ratios.
Triangle BCDBCD is an isosceles right triangle with right angle at CC and hypotenuse BD=12BD = 12. The legs are equal, so BC=CDBC = CD.
In a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle, hypotenuse = leg×2\text{leg} \times \sqrt{2}.
3
Solve for leg BCBC and rationalize the denominator.
BC=BD2=122=1222=62BC = \frac{BD}{\sqrt{2}} = \frac{12}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2}.
Dividing the hypotenuse by 2\sqrt{2} yields the leg length in standard simplified radical form.

Key Concept

Applying 30∘−60∘−90∘30^\circ-60^\circ-90^\circ and 45∘−45∘−90∘45^\circ-45^\circ-90^\circ special right triangle side ratio rules across multi-step figures.
Estimated Time:1m 15s
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Topics

Pythagorean Theorem and Special Right Triangles

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