Question

question 9 of 25 in △jkl, m∠j = 90°, m∠k = 30°, and m∠l = 60°. which of the following statements about △jkl are true? check all that apply. a. jk = 2·jl b. kl = √3·jl c. kl = 2·jl d. jk = √3·jl e. jl = √3/2·kl f. jk = √3/2·kl

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratios

In a right - triangle with angles 30°, 60°, and 90°, if the side opposite the 30° angle is \(a\), the side opposite the 60° angle is \(\sqrt{3}a\), and the hypotenuse is \(2a\). In \(\triangle{JKL}\), \(\angle{K} = 30^{\circ}\), \(\angle{L}=60^{\circ}\), and \(\angle{J} = 90^{\circ}\). Let \(JL\) be the side opposite the 30° angle (\(\angle{K}\)), \(JK\) be the side opposite the 60° angle, and \(KL\) be the hypotenuse.

Step2: Determine side - length relationships

We know that \(KL = 2\cdot JL\) (since the hypotenuse is twice the side opposite the 30° angle), and \(JK=\sqrt{3}\cdot JL\) (side opposite 60° is \(\sqrt{3}\) times the side opposite 30°). Also, \(JL=\frac{1}{2}KL\) and \(JK = \frac{\sqrt{3}}{2}KL\).

Answer:

A. \(JK = 2\cdot JL\) (False)
B. \(KL=\sqrt{3}\cdot JL\) (False)
C. \(KL = 2\cdot JL\) (True)
D. \(JK=\sqrt{3}\cdot JL\) (True)
E. \(JL=\frac{\sqrt{3}}{2}\cdot KL\) (False)
F. \(JK=\frac{\sqrt{3}}{2}\cdot KL\) (True)