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Circles with centres A, B and C touch each other externally. If AB = 36, BC = 32, CA = 30, then find the radii of each circle.

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#### Solution

**Given:** AB = 36, BC = 32, CA = 30

**To Find:** Radii of each circle.

**Solution:**

Let x, y, z be the radii of the circles with centers A, B, C respectively.

∴ AP = RA = x, PB = BQ = y, CR = QC = z

AB = AP + PB ....[A–P–B]

∴ 36 = x + y ...(ii) [From (i) and given]

BC = BQ + QC ....[B – Q – C]

∴ 32 = y + z .....(iii) [From (i) and given]

CA = CR + RA ......[C – R – A]

∴ 30 = z + x ......(iv) [From (i) and given]

Now,

AB + BC + CA = 36 + 32 + 30

∴ (AP + PB) + (BQ + QC) + (CR + RA) = 98 ......[A–P–B, B–Q–C, C–R–A]

∴ (x + y) + (y + z) + (z + x) = 98 ......[From (i)]

∴ 2x + 2y + 2z = 98

∴ 2(x + y + z) = 98

∴ x + y + z = `98/2`

∴ x + y + z = 49

∴ (x + y) + z = 49 .....[From (ii)]

∴ 36 + z = 49

∴ z = 49 – 36

∴ z = 13 ......(v)

y + z = 32 ......[From (iii)]

∴ y + 13 = 32 ......[From (v)]

∴ y = 32 – 13

∴ y = 19

z + x = 30 .....[From (iv)]

∴ 13 + x = 30 .....[From (v)]

∴ x = 30 – 13

∴ x = 17

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