Question

Secants L J and L M intersect and form an angle at point L. Solve for x.

Answer 1

x = 14

Explanation:

Assume your diagram is like the one below.

The intersecting secant angles theorem states, "When two secants intersect outside a circle, the measure of the angle formed is one-half the difference between the far and the near arcs."

For your diagram, that means

`\begin{array}{rcl}m\angle L &=&(1)/(2) \left(m \widehat {JM} - m\widehat {PQ}\right)\\\\(3x + 13)^(\circ)& = &(1)/(2) \left[(8x + 48)^(\circ) - (5x - 20)^(\circ)\right]\\\\3x + 13& = &(1)/(2)(8x + 48 - 5x + 20)\\\\3x + 13& = &(1)/(2)(3x + 68)\\\\6x + 26 & = & 3x + 68\\6x & = & 3x + 42\\3x & = & 42\\x & = & \mathbf{14}\\\end{array}`

Check:

`\begin{array}{rcl}(3*14 + 13) & = &(1)/(2) \left[(8*14 + 48)^(\circ) - (5*14 - 20)^(\circ)\right]\\\\42 + 13& = &(1)/(2)(112 + 48 - 70 + 20)\\\\55& = &(1)/(2)(110)\\\\55 & = & 55\\\end{array}`

It checks.