Question

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

Answer 1

(i) Perimeter of $△$ABC = AB + BC + AC
= AD + BD + BE + EC + CF + FA
= 5 + 5+ 5 + 5 + 5 + 5 = 30 cm

(ii) Since all radii are the same, AD = BD
Hence, D is the midpoint of AB.
Similarly, F and E are the midpoints of AC and BC.

i.e., AC = AF + FC = 5 + 5 = 10 cm
∴ DE = $\frac{1}{2}\mathrm{AC}=\frac{1}{2}\times 10=5\mathrm{cm}$ (By midpoint theorem)