Mathematics
In the figure (ii) given below, ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6 cm (not drawn to scale). Three circles are drawn touching each other with vertices A, B and C as their centres. Find the radii of the three circles.
Answer
Given, AB = 10 cm, BC = 8 cm, AC = 6 cm
Let the radius of the circles with centre A, B and C be x cm, y cm and z cm.
From figure,
AB = 10 cm.
⇒ x + y = 10 cm ….(i)
BC = 8 cm.
⇒ y + z = 8 cm ….(ii)
AC = 6 cm.
⇒ x + z = 6 cm …..(iii)
Adding eqn. (i), (ii) and (iii) we get,
⇒ x + y + y + z + x + z = (10 + 8 + 6) cm
⇒ 2x + 2y + 2z = 24 cm
⇒ 2(x + y + z) = 24 cm
⇒ x + y + z = 12 cm …..(iv)
Now, (iv) - (i) we get,
⇒ x + y + z - (x + y) = (12 - 10) cm
⇒ x + y + z - x - y = 2 cm
⇒ z = 2 cm.
Also, by (iv) - (ii) we get,
⇒ x + y + z - (y + z) = (12 - 8) cm
⇒ x + y + z - y - z = 4 cm
⇒ x = 4 cm
Also, by (iv) - (iii) we get,
⇒ x + y + z - (x + z) = (12 - 6) cm
⇒ x + y + z - x - z = 6 cm
⇒ y = 6 cm.
Hence, the radius of the circle with centre A, B and C are 4 cm, 6 cm and 2 cm.
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