Given:
A figure of a circle and two secants on the circle from the outside of the circle.
To find:
The measure of angle KLM.
Solution:
According to the intersecting secant theorem, if two secant of a circle intersect each other outside the circle, then the angle formed on the intersection is half of the difference between the intercepted arcs.
Using intersecting secant theorem, we get
[tex]\angle KLM=\dfrac{1}{2}(Arc(JON)-Arc(KM))[/tex]
[tex](3x-4)=\dfrac{1}{2}(271-(x+6))[/tex]
[tex](3x-4)=\dfrac{1}{2}(271-x-6)[/tex]
Multiply both sides by 2.
[tex]6x-8=265-x[/tex]
Isolate the variable x.
[tex]6x+x=265+8[/tex]
[tex]7x=273[/tex]
Divide both sides by 7.
[tex]x=\dfrac{273}{7}[/tex]
[tex]x=39[/tex]
Now,
[tex]\angle KLM=(3x-4)^\circ[/tex]
[tex]\angle KLM=(3(39)-4)^\circ[/tex]
[tex]\angle KLM=(117-4)^\circ[/tex]
[tex]\angle KLM=113^\circ[/tex]
Therefore, the measure of angle KLM is 113 degrees.
