Two circles with centers C₁ and C₂ have radii r₁ = 8 cm and r₂ = 7 cm, respectively. The distance between their centers is 17 cm. A common internal tangent touches the circles at points P₁ and P₂. What is the length of the segment P₁P₂?
8 cm
A common internal (transverse) tangent crosses the line joining the two centres, touching each circle on opposite sides. Its length is given by \(P_1P_2 = \sqrt{d^2 - (r_1 + r_2)^2}\), where \(d\) is the distance between centres.
The sum \(r_1 + r_2\) is used (not the difference) because an internal tangent reaches across between the circles; the difference \(r_1 - r_2\) would give the external tangent instead.
Here \(d = 17\text{ cm}\), \(r_1 = 8\text{ cm}\) and \(r_2 = 7\text{ cm}\), so \(r_1 + r_2 = 15\text{ cm}\).
Substitute: \(P_1P_2 = \sqrt{17^2 - 15^2} = \sqrt{289 - 225}\).
Therefore \(P_1P_2 = \sqrt{64} = 8\text{ cm}\).
Key concept: internal tangent uses \(\sqrt{d^2 - (r_1 + r_2)^2}\); the length exists only when \(d \gt r_1 + r_2\), which holds here. Hence the length of segment P₁P₂ is 8 cm.
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