Question

Rose watches a circular ornament that has 6 equally spaced stones. Identify x.

The figure shows a circle with tangent O B and secant O C. The circle is divided into 6 equal parts. Points A, B, C lie on the circle. Point A lies on secant O C. Intercepted arc A B measures 1 part. Intercepted arc B C measures 3 parts. The angle between the secant and the tangent measures x degrees. The tangent and the secant intersect in the exterior of the circle.

Answer 1

x = 60°

Explanation:

It is given that the stones are equally spaced. As the circle is divided into 6 parts by 6 split points and there are 360∘ in the circle, each part equals 360∘/6=60∘

The figure shows the same circle as in the beginning of the task. Arc A B measures 60 degrees. Arc B C measures 180 degrees.

So,

mAB◠=60∘

mBC◠=3(60∘)

=180∘

If a tangent and a secant intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs.

x=12(mBC◠−mAB◠)

Substitute the known values into the formula.

x=12(180∘−60∘)

=12(120∘)

=60∘

Therefore, x=60∘

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