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Difficulty: HardPythagorean Theorem and Special Right Triangles

An equilateral triangle ABCABC has a side length of 88 inches. Point DD lies on side BCBC such that the distance from BB to DD is 33 inches. What is the length, in inches, of the segment ADAD?

Answer: 7 inches

Answer

The length of segment ADAD is 77 inches.
Dropping altitude AMAM from AA to BCBC divides the equilateral triangle into two 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangles. Since MM is the midpoint of BCBC, BM=4BM = 4 inches. In △ABM\triangle ABM, the hypotenuse is 88 and the shorter leg is 44, so the altitude AM=43AM = 4\sqrt{3} inches. Since BD=3BD = 3 inches, the segment DMDM has length BM−BD=4−3=1BM - BD = 4 - 3 = 1 inch. Applying the Pythagorean Theorem to right triangle △ADM\triangle ADM gives AD2=AM2+DM2=(43)2+12=48+1=49AD^2 = AM^2 + DM^2 = (4\sqrt{3})^2 + 1^2 = 48 + 1 = 49, which simplifies to AD=7AD = 7 inches.

Step-by-Step Solution

1
Find the midpoint of side BCBC by dropping altitude AMAM.
BM=4BM = 4 inches
In an equilateral triangle, the altitude to a side bisects that side.
2
Calculate the length of the altitude AMAM.
AM=43AM = 4\sqrt{3} inches
The altitude forms a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle with the hypotenuse of 88 inches, making the altitude length equal to 8×32=438 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}.
3
Determine the length of the segment DMDM.
DM=1DM = 1 inch
Since DD is 33 inches from BB and MM is 44 inches from BB, the remaining distance is 4−3=14 - 3 = 1.
4
Apply the Pythagorean Theorem on right triangle △ADM\triangle ADM to find ADAD.
AD=7AD = 7 inches
The hypotenuse squared is the sum of the squares of the legs: AD2=(43)2+12=48+1=49AD^2 = (4\sqrt{3})^2 + 1^2 = 48 + 1 = 49, which gives AD=7AD = 7.

Key Concept

Using the altitude of an equilateral triangle to create special right triangles and applying the Pythagorean Theorem.
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Topics

Pythagorean Theorem and Special Right Triangles

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