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.
[tex]$\Rightarrow 51^{\circ}=\frac{1}{2}(160^{\circ} -x^{\circ})[/tex]
Multiply by 2 on both sides of the equation.
[tex]$\Rightarrow 51^{\circ}\times 2=2\times\frac{1}{2}(160^{\circ} -x^{\circ})[/tex]
[tex]$\Rightarrow 102^{\circ}=160^{\circ} -x^{\circ}[/tex]
Subtract 160° on both sides of the equation.
[tex]$\Rightarrow 102^{\circ}-160^{\circ}=160^{\circ} -x^{\circ}-160^{\circ}[/tex]
[tex]$\Rightarrow -58^{\circ}=-x^{\circ}[/tex]
Multiply by (–1) on both sides of the equation.
[tex]$\Rightarrow -58^{\circ}\times(-1)=-x^{\circ}\times(-1)[/tex]
[tex]$\Rightarrow x^{\circ}=58^{\circ}[/tex]
Hence the value of x is 58°.
