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प्रश्न
Circles with centres A, B and C touch each other externally. If AB = 3 cm, BC = 3 cm, CA = 4 cm, then find the radii of each circle.
उत्तर
Let AP = AR = x, BP = BQ = y, CQ = CR = z .....[Radii of the same circle]
AP + BP = AB …[A – P – B]
∴ x + y = 3 …(i)
BQ + CQ = BC …[B – Q – C]
∴ y + z = 3 …(ii)
AR + CR = AC …[A – R – C]
∴ x + z = 4 …(iii)]
Adding equations (i), (ii) and (iii), we get
x + y + y + z + x + z = 3 + 3 + 4
∴ 2x + 2y + 2z = 10
∴ 2(x + y + z) = 10
∴ x + y + z = 5 …(iv)
Substituting equation (i) in equation (iv), we get 3 + z = 5
∴ z = 5 − 3
∴ z = 2 cm
Substituting equation (ii) in equation (iv), we get x + 3 = 5
∴ x = 5 − 3
∴ x = 2 cm
Substituting equation (iii) in equation (iv), we get
y + 4 = 5
∴ y = 5 − 4
∴ y = 1 cm
∴ The radii of circles with centres A, B, C are 2 cm, 1 cm and 2 cm respectively.
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