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In the circle above with center C, diameter AD = 18. Which

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In the circle above with center C, diameter AD = 18. Which

by BTGmoderatorDC » Sat Mar 23, 2019 8:47 pm

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In the circle above with center C, diameter AD = 18. Which of the following must be true?

I. The length of minor arc BD is greater than 10.

II. Angle ABD measures 90 degrees.

III. Triangle BCD is equilateral.

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

OA D

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by deloitte247 » Sun Mar 31, 2019 12:31 am
If AD=18, then AC=9 and CD=9.
(i) Length of the minor arc BD is greater than 10.
$$Length\ of\ arc\ BD=2\pi r\ \frac{C}{360^0}$$
Where C = central angle and radius r = radius.
BC=CD= radius of the circle
Radius = 9 and angle c = 60 degrees.
$$Length\ of\ arc\ BD=2\pi\cdot9\cdot\ \frac{60}{360^0}=\ 3\pi\ =\ 9.42$$

(ii) In triangle ABD, angle B =90 degrees because triangle ABD is inside semicircle with one side AD as the diameter.

(iii) Triangle BCD is equilateral from the diagram shown in question. Angle A=30 degrees from (ii) above and Angle b=90 degrees. Hence, triangle ABD is in 30, 90 and 60 degrees respectively.
So, angle D=60 degrees; this thus make angle C = 60 degrees.
$$Therefore,\ \triangle BCD=equilateral\ triangle\ -\ 60^0,60^0,60^0$$

Thus, (i) is false. Only (ii) and (iii) are true. So, the correct answer is option D

Hope this helps?
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