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Difficulty: EasyPythagorean Theorem and Special Right Triangles

In right triangle DEFDEF, the measure of ∠E\angle E is 90∘90^\circ and the measure of ∠D\angle D is 60∘60^\circ. If the hypotenuse DFDF has a length of 1414 centimeters, what is the length, in centimeters, of the segment DEDE?

  1. A
    727\sqrt{2}
  2. B
    737\sqrt{3}
  3. 77Answer
  4. D
    14314\sqrt{3}
  5. E
    2828

Answer

The length of the segment DEDE is 77 centimeters.
The correct option is the one with the value 77. In right triangle DEFDEF, the angles are 90∘90^\circ, 60∘60^\circ, and 30∘30^\circ, making it a special 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangle. The side DEDE is opposite the 30∘30^\circ angle (the shorter leg). By the properties of a 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle, the shorter leg is half the length of the hypotenuse. Thus, DE=14/2=7DE = 14 / 2 = 7 centimeters.

Step-by-Step Solution

1
Determine the measure of the third angle, ∠F\angle F.
∠F=30∘\angle F = 30^\circ
The sum of angles in a triangle is 180∘180^\circ. Since ∠E=90∘\angle E = 90^\circ and ∠D=60∘\angle D = 60^\circ, we calculate ∠F=180∘−90∘−60∘=30∘\angle F = 180^\circ - 90^\circ - 60^\circ = 30^\circ.
2
Identify the relationship between the sides of the 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle.
DEDE is the shorter leg, opposite ∠F\angle F (30∘30^\circ).
The side opposite the 30∘30^\circ angle is the shorter leg, which is half the length of the hypotenuse.
3
Calculate the length of DEDE.
DE=7DE = 7 centimeters
Since the hypotenuse DF=14DF = 14 centimeters, the shorter leg DEDE is 14/2=714 / 2 = 7 centimeters.

Key Concept

In a 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangle, the lengths of the sides are in the ratio 1:3:21 : \sqrt{3} : 2. The shorter leg (opposite the 30∘30^\circ angle) is half the length of the hypotenuse.
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Topics

Pythagorean Theorem and Special Right Triangles

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