Question

Three congruent circles with centres $A,B$ and $C$ with radius $5cm$ each, touch each other in point $D,E,F$ as shown in figure.
(i) What is the perimeter of $\Delta ABC$?
(ii) What is the length of side $DF$ of $\Delta DEF$?

Answer 1

Given $A,B,C$ are congruent circles with radius $=5cm$ each.

(i) For the perimeter of $\Delta ABC$

$\Rightarrow AB=(5+5)cm=10cm$

$BC=(5+5)cm=10cm$

$CA=(5+5)cm=10cm$

$\therefore$ Perimeter of $\Delta ABC$$=AB+BC+CA=(10+10+10)cm$$=30cm$

(ii) Consider $\Delta ADF$

$\because$ From triangle $ABC$, all sides are equal.

$\therefore \angle A={60}^o$ [equilateral triangle]

$AF=AD$

$\therefore \angle F=\angle D=x\left(let\right)$ [If $2$ sides are equal then their opposite angles are also equal]

$\therefore x+x+60=180$

$\Rightarrow 2x=120$

$\Rightarrow x={60}^o$

$\therefore$ $ADF$ is a equilateral triangle

$\therefore DF=AD=5cm$.

Hence the length of $DF$ is $5cm$