Question

A circle is shown. A secant and a tangent intersect at a common point outside of the circle to form an angle that measure 51 degrees. The measure of the first arc formed is x degrees and the measure of the second arc formed is 160 degrees. What is the value of x? x =

Answer 1

Answer:

58

Step-by-step explanation:

Answer 2

The value of x is 58°.

Solution:

The measure of the first arc formed = x°

The measure of the second arc formed = 160°

Angle formed between tangent and secant = 51°

Theorem:

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

$\Rightarrow 51^{\circ}=\frac{1}{2}(160^{\circ} -x^{\circ})$

Multiply by 2 on both sides of the equation.

$\Rightarrow 51^{\circ}\times 2=2\times\frac{1}{2}(160^{\circ} -x^{\circ})$

$\Rightarrow 102^{\circ}=160^{\circ} -x^{\circ}$

Subtract 160° on both sides of the equation.

$\Rightarrow 102^{\circ}-160^{\circ}=160^{\circ} -x^{\circ}-160^{\circ}$

$\Rightarrow -58^{\circ}=-x^{\circ}$

Multiply by (–1) on both sides of the equation.

$\Rightarrow -58^{\circ}\times(-1)=-x^{\circ}\times(-1)$

$\Rightarrow x^{\circ}=58^{\circ}$

Hence the value of x is 58°.