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प्रश्न
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.
उत्तर
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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