Question

Given that line AB is tangent to the circle, find m

Answer 1

Solution:

Given the figure below:

To solve for m∠CAB, we use the chord-tangent theorem which states that when a chord and a tangent intersect at a point, it makes angles that are half the intercepted arc.

Thus,

$m\angle CAB=\frac{1}{2}\times arc\text{ CDB}$

where

$\begin{gathered} m\angle CAB=(4x+37)\degree \\ arc\text{ CDB=\lparen9x+53\rparen}\degree \end{gathered}$

By substituting these values into the above equation, we have

$4x+37=\frac{1}{2}(9x+53)$

Multiplying through by 2, we have

$\begin{gathered} 2(4x+37)=(9x+53) \\ open\text{ parentheses,} \\ 8x+74=9x+53 \end{gathered}$

Collect like terms,

$\begin{gathered} 8x-9x=53-74 \\ \Rightarrow-x=-21 \\ divide\text{ both sides by -1} \\ -\frac{x}{-1}=-\frac{21}{-1} \\ \Rightarrow x=21 \end{gathered}$

Recall that

$\begin{gathered} m\operatorname{\angle}CAB=(4x+37)\operatorname{\degree} \\ where \\ x=21 \\ thus, \\ m\operatorname{\angle}CAB=4(21)+37 \\ =84+37 \\ \Rightarrow m\operatorname{\angle}CAB=121\degree \end{gathered}$

Hence, the measure of the angle CAB is

$121\degree$