Q3. - Geometry

Question

$\overline{JT}$ is tangent to $\odot$ $O$ at $T$. Complete.

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

Step-by-Step Solution

Answer

If $m\angle TOJ=60$ and $OT=6$, then $JO=12$.

Step 1. Given information.

The statement here given is,

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

The figure:

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

Step 2. Concept Used.

According to theorem $9·2$,

 Tangent of the circle is perpendicular to the radius of the circle.

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

Step 3. 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 $\Delta OTJ$.

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

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

Step 4. Now, perform cross multiplication.

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

Therefore, the complete statement is,

If $m\angle TOJ=60$ and $OT=6$, then $JO=12$.