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Difficulty: HardRight Triangles and the Pythagorean Theorem

In triangle ABCABC, angle CC is a right angle. The length of side ACAC is 1515 and the length of side BCBC is 88. Point DD lies on side ACAC such that CD=6CD = 6. A line segment DEDE is drawn perpendicular to side ACAC such that point EE lies on side ABAB. What is the length of segment BEBE?

  1. A
    7.8
  2. 6.8Answer
  3. C
    10.2
  4. D
    9.1

Answer

The correct answer is 6.8.
The length of the hypotenuse ABAB is calculated using the Pythagorean theorem: AB=152+82=17AB = \sqrt{15^2 + 8^2} = 17. The length of segment ADAD is 15−6=915 - 6 = 9. Because segment DEDE is perpendicular to ACAC, the right triangle ADEADE shares angle AA with right triangle ACBACB, making them similar. Using the ratio of corresponding sides, ADAC=AEAB\frac{AD}{AC} = \frac{AE}{AB}, we get 915=AE17\frac{9}{15} = \frac{AE}{17}, which simplifies to AE=10.2AE = 10.2. Finally, subtracting AEAE from the total hypotenuse length gives BE=17−10.2=6.8BE = 17 - 10.2 = 6.8.

Step-by-Step Solution

1
Find the length of the hypotenuse ABAB using the Pythagorean theorem.
AB=17AB = 17
For a right triangle with legs AC=15AC = 15 and BC=8BC = 8, the hypotenuse ABAB is given by 152+82=225+64=289=17\sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17.
2
Calculate the length of segment ADAD.
AD=9AD = 9
Since point DD lies on side ACAC and the total length of ACAC is 1515, the length of ADAD is AC−CD=15−6=9AC - CD = 15 - 6 = 9.
3
Identify the similarity relationship between triangle ADEADE and triangle ACBACB.
△ADE∼△ACB\triangle ADE \sim \triangle ACB
Both triangles share angle AA, and both have a right angle (∠ADE=∠ACB=90∘\angle ADE = \angle ACB = 90^\circ). Therefore, by Angle-Angle similarity, the triangles are similar.
4
Use the similarity ratio to find the length of segment AEAE.
AE=10.2AE = 10.2
The ratio of corresponding sides gives ADAC=AEAB\frac{AD}{AC} = \frac{AE}{AB}, which simplifies to 915=AE17\frac{9}{15} = \frac{AE}{17}, or 0.6=AE170.6 = \frac{AE}{17}. Solving for AEAE gives AE=17×0.6=10.2AE = 17 \times 0.6 = 10.2.
5
Subtract AEAE from the total hypotenuse ABAB to find the length of segment BEBE.
BE=6.8BE = 6.8
The length of segment BEBE is the difference between the total hypotenuse ABAB and the segment AEAE, which is 17−10.2=6.817 - 10.2 = 6.8.

Key Concept

Applying the Pythagorean theorem and similar right triangles to find missing lengths on a hypotenuse.
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Topics

Right Triangles and the Pythagorean Theorem

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