Question

In circle O, chord AOB is drawn. Point C is located on the exterior of the circle with tangent CB and secant CDA drawn.Itisknownthat CD2 and DA6.

(a) Determine the length of BC .
(b) What is the length of the radius of circle O in simplest radical form.
(c) Determine the measure of C . Hint - consider right triangle trigonometry instead of circle theorems.
REASONING
(d) Find the measure of AD . Show how you arrived at your answer.

Answer 1

(a) 4

(b) 2√3

(c) 60°

(d) 120°

Explanation:

(a) The relationship between tangents and secants is ...

CB^2 = CD·CA

Filling in the given values, we find ...

CB^2 = 2·(2+6) = 16

CB = √16 = 4

The length of BC is 4 units.

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(b) Triangle ABC is a right triangle, so the sides of it satisfy the Pythagorean theorem.

CA^2 = CB^2 +AB^2

8^2 = 16 +AB^2

AB = √48 = 4√3

The radius is half the length of AB, so the radius is 2√3.

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(c) The measure of angle C can be determined from the cosine relation:

cos(C) = CB/CA = 4/8 = 1/2

C = arccos(1/2) = 60°

The measure of angle C is 60°.

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(d) Arc AD is intercepted by angle ABD, which has the same measure as angle C. Hence the measure of arc AD is twice the measure of angle C.

The measure of arc AD is 120°.