Question

A tangent is drawn from an object in space to Earth, and a secant is drawn from thesame point through the center of Earth.What is mZS?37°16°106°BO32°8°

Answer 1

The angle S can be obtained as follows:

Step 1: Make a well labelled sketch of the diagram, as below:

As seen above, the center of the circle is labeled O, and the Tangent to the circle is labeled A. Also, because the angle between a tangent to a circle and the radius of the circle is a right angle (90 degrees), that angle has also been added to the sketch.

Step 2: Consider the triangle SAO, we know that the angle at A is 90 degrees (as explained perviously). Now the angle at O is obtained as follows:

Rule: Sum of angles on a straight line is 180 degrees

Thus, we have:

`\measuredangle O+106^o=180^o`

and:

`\begin{gathered} \measuredangle O+106^o=180^o \\ \Rightarrow\measuredangle O=180^o-106^o \\ \Rightarrow\measuredangle O=74^o \end{gathered}`

Now, we apply the following rule to obtain the value of the measure of angle S

Rule: Sum of angles in a triangle is 180 degrees

Considering triangle SAO, we have:

`\measuredangle S+\measuredangle A+\measuredangle O=180^o`

Thus:

`\begin{gathered} \measuredangle S+\measuredangle A+\measuredangle O=180^o \\ \Rightarrow\measuredangle S+90^o+74^o=180^o \\ \Rightarrow\measuredangle S+=180^o-90^o-74^o \\ \Rightarrow\measuredangle S=180^o-164^o \\ \Rightarrow\measuredangle S=16^o \end{gathered}`

Thus the value of the measure of angle S is 16 degrees