Chapter 9.2, Problem 3WE

Solution

To completer: the given statement.

If $m\angle TOJ=60$ and $OT=6$ , then $JO=\underset{\_}{12}$

Given:

The statement:

If $m\angle TOJ=60$ and $OT=6$ , then $JO=\_\_\_\_$

The figure:

  $JT$ is a tangent to circle with centre O at T .

Concept Used:

Theorem 9-2: tangent of the circle is perpendicular to the radius of the circle.

  $\cos A=\frac{\text{leg adjacent to}\angle A}{\text{hypotenuse}}$

Explanation:

Consider the given figure:

It is given that $JT$ is a tangent to circle with centre O at T .

So, by theorem 9-2, it can be said that $\overline{JT}$ is perpendicular to $\overline{OT}$ .

That is,

  $m\angle OTJ=90$

Consider the right triangle $△OTJ$ .

Now, it is given that $m\angle TOJ=60$ , so

  $\begin{array}{c}\cos \left(TOJ\right)=\frac{\text{leg adjacent to}\angle TOJ}{\text{hypotenuse}}\\ \cos 60°=\frac{OT}{JO}\\ \frac{1}{2}=\frac{6}{JO}\end{array}$

Now perform cross multiplication,

  $\begin{array}{c}JO=6\times 2\\ JO=12\end{array}$

Hence, the complete statement is:

If $m\angle TOJ=60$ and $OT=6$ , then $JO=\underset{\_}{12}$