First find the measure of the angle formed when two tangents intersect in the exterior of a circle.
7^(∘)
Practice makes perfect
A signal from a tower at sea level follows a ray. The endpoint of the ray is tangent to Earth. Also, the measure of the angle formed by the center of Earth, the top of the tower, and a tangent is given as 86.5^(∘).
We want to determine the measure of the arc intercepted by the two tangents. To do so we will first find the measure of the angle constructed by the tangents. Let's start by labeling the vertices of the given diagram
T_1C and T_2C are the radius of Earth, so they are congruent. Also, SC is the hypotenuse of the both △ST_1C and △ST_2C. To check whether T_1S and T_2S are congruent as well, we will use the following theorem.
If two segments from the same exterior point are tangent to a circle, then they are congruent.
By the theorem the segments T_1S and T_2S are congruent. Therefore, by Side-Side-Side Congruence Theorem △ST_1C and △ST_2C are congruent. With this information, ∠T_1SC and ∠T_2SC are congruent.
m∠T_1SC = m∠T_2SC
m∠T_1SC = 86.5^(∘) ⇒ m∠T_2SC=86.5
We can find the measure of ∠T_1ST_2.
Next we will find the measure of the arc intercepted by the two tangents. To find it we will use the theorem about the measure of an angle formed by two tangents.
If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
To use the theorem, let's identify the measures of the intercepted arcs.
The measures of the intercepted arcs are mT_1T_2 and 360-mT_1T_2. With this information we can use the theorem.
