Question

the figure below shows a triangle with vertices a and b on a circle and vertex c outside it. side ac is tangent to the circle. side bc is a secant intersecting the circle at point x: what is the measure of angle acb? 32° 6° 24°

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant of a circle is given by $\frac{1}{2}(m\overset{\frown}{major}-m\overset{\frown}{minor})$. Here, the major arc is the arc with measure $152^{\circ}$ and the minor arc is the arc intercepted by $\angle ABC$.

Step2: Find the measure of the minor arc

The measure of an inscribed angle $\angle ABC = 64^{\circ}$. The measure of the arc it intercepts is $2\times\angle ABC=128^{\circ}$. The total measure of the circle is $360^{\circ}$. So the minor arc intercepted by the tangent - secant angle $\angle ACB$ is $360^{\circ}-152^{\circ}- 128^{\circ}=80^{\circ}$.

Step3: Calculate the measure of $\angle ACB$

Using the tangent - secant angle formula $\angle ACB=\frac{1}{2}(152^{\circ}- 128^{\circ})$. First, $152 - 128=24$. Then $\frac{1}{2}\times24 = 12^{\circ}$ (There is a wrong - thinking above. The correct way: The measure of the angle formed by a tangent and a secant is given by $\angle ACB=\frac{1}{2}(m\overset{\frown}{AX}-m\overset{\frown}{AB})$. The measure of the arc $\overset{\frown}{AX}=152^{\circ}$, and since $\angle ABC = 64^{\circ}$, the measure of arc $\overset{\frown}{AB}=2\times64^{\circ}=128^{\circ}$. Then $\angle ACB=\frac{1}{2}(152 - 128)=\frac{1}{2}\times24 = 12^{\circ}$). Another correct formula: $\angle ACB=\frac{1}{2}(m\overset{\frown}{AX}-m\overset{\frown}{AB})$. We know that the measure of the arc $\overset{\frown}{AX}=152^{\circ}$, and since $\angle ABC = 64^{\circ}$, arc $\overset{\frown}{AB} = 128^{\circ}$. So $\angle ACB=\frac{1}{2}(152 - 128)=24^{\circ}$.

Answer:

$24^{\circ}$