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Difficulty: MediumPythagorean Theorem and Special Right Triangles

In the figure, point BB lies on the line segment ACAC, and segment BDBD is perpendicular to ACAC. Triangle ABDABD is a right triangle with hypotenuse AD=8AD = 8 and ∠ADB=30∘\angle ADB = 30^\circ. If the length of segment BCBC is 1111, what is the length of segment CDCD?

Answer: 13

Answer

The length of segment CDCD is 1313.
The length of segment CDCD is 1313. Since BDBD is perpendicular to segment ACAC at point BB, △ABD\triangle ABD and △DBC\triangle DBC are both right triangles. In the 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle △ABD\triangle ABD, the hypotenuse is AD=8AD = 8, so the longer leg opposite the 60∘60^\circ angle is BD=43BD = 4\sqrt{3}. In right triangle △DBC\triangle DBC, using the Pythagorean Theorem: CD2=BD2+BC2=(43)2+112=48+121=169CD^2 = BD^2 + BC^2 = (4\sqrt{3})^2 + 11^2 = 48 + 121 = 169. Taking the square root gives CD=13CD = 13.

Step-by-Step Solution

1
Find the length of the shared perpendicular segment BDBD using the properties of the special 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle △ABD\triangle ABD.
BD=43BD = 4\sqrt{3}
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle, the side opposite the 60∘60^\circ angle is 32\frac{\sqrt{3}}{2} times the hypotenuse.
2
Apply the Pythagorean Theorem to right triangle △DBC\triangle DBC to calculate the length of hypotenuse CDCD.
CD=13CD = 13
The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs (CD2=BD2+BC2CD^2 = BD^2 + BC^2).

Key Concept

Solving for unknown sides in adjacent right triangles by combining special right triangle ratios (30∘−60∘−90∘30^\circ-60^\circ-90^\circ) and the Pythagorean Theorem.
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Topics

Pythagorean Theorem and Special Right Triangles

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