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Difficulty: EasyPythagorean Theorem and Special Right Triangles

In right triangle PQRPQR, the measure of angle PP is 30∘30^\circ, the measure of angle QQ is 60∘60^\circ, and the length of the hypotenuse PRPR is 1212. Which of the following statements must be true? Select all that apply.

  1. The length of the shortest side is 66.Answer
  2. The length of the side opposite the 60∘60^\circ angle is 636\sqrt{3}.Answer
  3. C
    The length of the side opposite the 60∘60^\circ angle is 626\sqrt{2}.
  4. D
    The area of triangle PQRPQR is 36336\sqrt{3}.
  5. E
    The perimeter of triangle PQRPQR is 18318\sqrt{3}.

Answer

The statements confirming that the shortest side has length 66 and that the side opposite the 60∘60^\circ angle has length 636\sqrt{3} are both correct.
In any 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle, the sides opposite the 30∘30^\circ, 60∘60^\circ, and 90∘90^\circ angles are in the ratio 1:3:21 : \sqrt{3} : 2. With a hypotenuse of 1212, the side opposite 30∘30^\circ is 122=6\frac{12}{2} = 6, and the side opposite 60∘60^\circ is 636\sqrt{3}. Thus, both statements specifying these values are correct.

Step-by-Step Solution

1
Identify the side length ratio for a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle.
The side lengths follow the standard ratio 1:3:21 : \sqrt{3} : 2, corresponding to the sides opposite the 30∘30^\circ, 60∘60^\circ, and 90∘90^\circ angles respectively.
Special right triangle properties establish fixed proportional relationships between sides based on interior angles.
2
Calculate the length of the shortest side (opposite the 30∘30^\circ angle).
Shortest side = Hypotenuse2=122=6\frac{\text{Hypotenuse}}{2} = \frac{12}{2} = 6.
Since the hypotenuse corresponds to 2x=122x = 12, x=6x = 6.
3
Calculate the length of the longer leg (opposite the 60∘60^\circ angle).
Longer leg = x3=63x\sqrt{3} = 6\sqrt{3}.
The side opposite 60∘60^\circ is 3\sqrt{3} times the side opposite 30∘30^\circ.

Key Concept

Side length ratios of a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle (1:3:21 : \sqrt{3} : 2).
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Topics

Pythagorean Theorem and Special Right Triangles

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