Question

m<s=30°,mRS(arch)=84 degrees and RU is tangent to the circle at R. Find m<U. The figure is not drawn to scale.

A) 27°
B) 12°
C) 54°
D) 24°

Answer 1

B.
`12^(\circ)`

Explanation:

We have been given an image of a circle. We are asked to find the measure of angle U.

We have been given that measure of angle 's' is 30 degrees, so the measure of arc RT will be 2 times the measure of angle 's' as angle 's' is inscribed angle of arc RT.

`\text{Measure of arc RT}=2* 30^(\circ)`

`\text{Measure of arc RT}=60^(\circ)`

We know that the measure of angle formed by intersecting secant and tangent outside a circle is half the difference of intercepted arcs.

Using Secant-tangent theorem, we can set an equation to find the measure of angle U as:

`m\angle U=(1)/(2)* (\text{Measure of arc RS}-\text{Measure of arc RT})`

Substituting the given values in above equation we will get,

`m\angle U=(1)/(2)*(84^(\circ)-60^(\circ))`

`m\angle U=(1)/(2)*(24^(\circ))`

`m\angle U=12^(\circ)`

Therefore, the measure of angle U is 12 degrees and option B is the correct choice.

Answer 2

The correct answer among the choices provided is option B. Resulting angle m < U is equivalent to 12 °.

The first step is determine ∠SRU,
180° - 84° = 96°
98°/2 = 48°
∠SRU = 48° + 90° = 138°

Then angle U from the triangle was determined using this formula,
∠U = 180° - ∠SRU - ∠S
= 180° - 138° - 30°
= 12°