Geometry

[swerve]

In the figure below, PQ is a diameter of circle O, PR = SQ, and ΔRST is equilateral. If the length of PQ is 2, what is the length of RT?

A. \(\frac{1}{2}\)
B. \(\frac{1}{\sqrt{3}}\)
C. \(\frac{\sqrt{3}}{2}\)
D. \(\frac{2}{\sqrt{3}}\)
E. \(\sqrt{3}\)

The OA is D

Source: Official Guide

[Jay@ManhattanReview]

In the figure below, PQ is a diameter of circle O, PR = SQ, and ΔRST is equilateral. If the length of PQ is 2, what is the length of RT?

A. \(\frac{1}{2}\)
B. \(\frac{1}{\sqrt{3}}\)
C. \(\frac{\sqrt{3}}{2}\)
D. \(\frac{2}{\sqrt{3}}\)
E. \(\sqrt{3}\)

The OA is D

Source: Official Guide

Join OT. Thus, OT = 1 , radius.

You would observe that ∆RTO is a rightangled triangle, with /_TOR = 90º; /_TRO = 60º (given that ∆RST is an equilateral triangle); thus, /_RTO = 30º. Thus, ∆RTO is a 30-60-90 rightangled triangle.

Note that the sides of a 30-60-90 rightangled triangle are in the ratio of 1 : √3 : 2, respectively. Thus, RT = (2/√3)*OT = (2/√3)*1 = 2/√3

The correct answer: D

Hope this helps!

-Jay
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