With the vertices of triangle PQR as the centers, three circles are described each touching the other two externally. If the sides of the triangle are 9cm, 7cm and 6cm. Find the radii of the circles.

Let PQ = 6 cm, QR = 9 cm and PR = 7 cm.

Suppose the radius of the circles with centres P, Q and R be z cm, x cm and y cm respectively.

When two circles touches each other externally, then the distance between their centres is equal to the sum of their radius.

∴ x + z = 6    ...(1)

x + y = 9       ...(2)

y + z = 7        ...(3)

Adding (1), (2) and (3), we get

2(x + y + z) = 6 + 9 + 7 = 22

∴ x + y + z = 11    ...(4)

From (1) and (4), we get

y + 6 = 11

∴ y = 11 – 7 = 5

From (2) and (4), we get

9 + z = 11

∴ z = 11 – 9 = 2

From (3) and (4), we get

x + 7 = 11

∴ x = 11 – 7 = 4

Thus, the radius of the circle with centre P, Q and R is 2 cm, 4 cm and 5 cm respectively.