Question

In the given figure, if $AB=3\text{units},AD=4\text{units},\angle HEG={130}^o$ and $DF=3\text{units},$ find the length of $BG.$
Here, C and E are the centers of respective circles.

• A. Insufficient data.
• B. Between 2 to 4 units.
• C. Between 5 to 7 units.
• D. Between 7 to 10 units.

Answer 1

The correct option is D Between 7 to 10 units.

The given figure is as follows:

From the left circle, $△ABC$ is a right triangle as the angle subtended by the diameter BD to A on the circle is equal to ${90}^o.$
Hence, $\angle A={90}^o,$ and BD is the hypotenuse.

Given: $AB=3\text{units},AD=4\text{units}$
$\therefore BD=\sqrt{3^2+4^2}=5\text{units}$

We Know: The angle subtended by the arc CG at the centre of a cirlce must be double the angle subtended by the arc CG at point F on the circle.
$\Rightarrow \angle CFG=\frac{\angle CEG}{2}=\frac{{130}^o}{2}={65}^o$

From the figure, $\angle DGF={65}^o=\angle DFG$
$\therefore$ For $△DGF,$ the opposite side of the mentioned angles should be equal.

Hence, $DG=DF=3\text{units}$

$\therefore BG=BD+DG=5+3=8\text{units}$