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Difficulty: HardPythagorean Theorem and Special Right Triangles

An equilateral triangle ABCABC has a side length of 1212 inches. An altitude ADAD is drawn from vertex AA to the side BCBC. A point PP lies on the segment ADAD such that △BPC\triangle BPC is a right triangle with a right angle at PP. What is the length, in inches, of the segment APAP?

  1. A
    63−326\sqrt{3} - 3\sqrt{2}
  2. B
    63−626\sqrt{3} - 6\sqrt{2}
  3. 63−66\sqrt{3} - 6Answer
  4. D
    63+66\sqrt{3} + 6
  5. E
    123−612\sqrt{3} - 6

Answer

The correct answer is 63−66\sqrt{3} - 6 inches.
The correct answer is 63−66\sqrt{3} - 6. Since the side length of the equilateral triangle is 1212, the altitude ADAD splits it into two 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangles with base BD=6BD = 6 and altitude AD=63AD = 6\sqrt{3}. The right triangle △BPC\triangle BPC has ∠BPC=90∘\angle BPC = 90^\circ and PB=PCPB = PC, making it an isosceles right triangle. The altitude PDPD splits △BPC\triangle BPC into two 45∘45^\circ-45∘45^\circ-90∘90^\circ right triangles, so PD=BD=6PD = BD = 6. The length of APAP is found by subtracting PDPD from ADAD, yielding 63−66\sqrt{3} - 6.

Step-by-Step Solution

1
Find the length of the altitude ADAD using the properties of the 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle △ABD\triangle ABD.
The length of ADAD is 636\sqrt{3} inches.
Since △ABC\triangle ABC is equilateral with side length 1212 inches, the altitude ADAD bisects the base BCBC, making BD=6BD = 6 inches. The altitude splits the equilateral triangle into two 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangles. The length of the longer leg is the shorter leg multiplied by 3\sqrt{3}, which gives AD=63AD = 6\sqrt{3}.
2
Find the length of the segment PDPD using the properties of the 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle △PDB\triangle PDB.
The length of PDPD is 66 inches.
Since PP lies on the altitude ADAD, which is the axis of symmetry, △BPC\triangle BPC is an isosceles right triangle with ∠BPC=90∘\angle BPC = 90^\circ. The altitude PDPD is perpendicular to BCBC and bisects ∠BPC\angle BPC, forming two 45∘45^\circ-45∘45^\circ-90∘90^\circ right triangles: △PDB\triangle PDB and △PDC\triangle PDC. In a 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle, the two legs are congruent, so PD=BD=6PD = BD = 6.
3
Subtract the length of PDPD from the length of ADAD to find the length of segment APAP.
AP=63−6AP = 6\sqrt{3} - 6 inches.
Since point PP lies on segment ADAD, the length of APAP is the difference between the total altitude ADAD and the segment PDPD.

Key Concept

Properties of special right triangles (30-60-90 and 45-45-90) and their multi-step application in geometry.
Estimated Time:2m 0s
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Topics

Pythagorean Theorem and Special Right Triangles

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