Examkin

Question

Difficulty: MediumRight Triangles and the Pythagorean Theorem

In right triangle ABCABC, the measure of angle CC is 90∘90^\circ. If sin⁡A=513\sin A = \frac{5}{13} and the perimeter of the triangle is 6060, what is the length of the shortest side of triangle ABCABC?

  1. A
    5
  2. 10Answer
  3. C
    12
  4. D
    24

Answer

10
The trigonometric ratio sin⁡A=513\sin A = \frac{5}{13} gives the ratio of the side opposite to angle AA to the hypotenuse. Using the Pythagorean theorem, the adjacent side corresponds to the 5-12-13 right triangle ratio. The perimeter in terms of the ratio factor kk is 5k+12k+13k=30k5k + 12k + 13k = 30k. Setting 30k=6030k = 60 yields k=2k = 2. The shortest side of the triangle corresponds to the smallest ratio value, which is 5k5k. Multiplying 55 by the scale factor 22 yields a length of 1010.

Step-by-Step Solution

1
Use the definition of sine to set up the ratio of the side lengths.
Since sin⁡A=oppositehypotenuse=BCAB=513\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{5}{13}, let the side opposite to angle AA (BCBC) be 5k5k and the hypotenuse (ABAB) be 13k13k, where kk is a positive constant.
This relates the given trigonometric ratio to the side lengths of the right triangle.
2
Find the third side using the Pythagorean theorem or Pythagorean triples.
Using the Pythagorean theorem, AC2+BC2=AB2AC^2 + BC^2 = AB^2, we get AC2+(5k)2=(13k)2  ⟹  AC2+25k2=169k2  ⟹  AC2=144k2  ⟹  AC=12kAC^2 + (5k)^2 = (13k)^2 \implies AC^2 + 25k^2 = 169k^2 \implies AC^2 = 144k^2 \implies AC = 12k. This matches the 5-12-13 Pythagorean triple.
To find the perimeter, all three side lengths of the right triangle are needed.
3
Set up an equation for the perimeter and solve for kk.
The perimeter is the sum of all sides: 5k+12k+13k=30k5k + 12k + 13k = 30k. We are given that the perimeter is 6060. Thus, 30k=60  ⟹  k=230k = 60 \implies k = 2.
Using the given perimeter helps determine the exact side lengths.
4
Determine the length of the shortest side.
The side lengths are BC=5(2)=10BC = 5(2) = 10, AC=12(2)=24AC = 12(2) = 24, and AB=13(2)=26AB = 13(2) = 26. The shortest side is BCBC, which has a length of 1010.
The question asks for the length of the shortest side.

Key Concept

Using trigonometric ratios and the Pythagorean theorem to find side lengths of right triangles from perimeter.
Estimated Time:1m 30s
Rate this question

Topics

Right Triangles and the Pythagorean Theorem

More questions from this topic