From an external point P, two tangents PA and PB are drawn to a circle with center O. A secant passing through P cuts the circle at points C and D such that PC = 4 cm and PD = 8 cm. If the radius of the circle is 6 cm, find the length of the tangent PA.
\(4\sqrt{2}\) cm
Tangent–secant theorem: if from an external point P a tangent PA and a secant PCD are drawn to a circle, then \(PA^2 = PC \times PD\).
Substituting PC = 4 cm and PD = 8 cm: \(PA^2 = 4 \times 8 = 32\) cm2.
So \(PA = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\) cm. The radius (6 cm) is not needed for this result; the two tangents PA and PB are equal in length.
Check: \((4\sqrt{2})^2 = 16 \times 2 = 32 = 4 \times 8\); also the chord CD = 8 – 4 = 4 cm is shorter than the diameter 12 cm, so the given data are consistent.
Hence, \(4\sqrt{2}\) cm.
Similar Questions
Suppose C and D are centres of two circles of radii 16 cm and 9 cm respectively, and distance between the centres is 25 cm. Find the length of direct common tangent.
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Important Questions from Circles, Chords and Tangents
AB is a chord of a circle with centre O and P is any point on the circle. If ∠APB = 112°, then what is the measure of ∠OAB ?
The distance between the centres of two circles is 24 cm. If the radius of the two circles are 4 cm and 8 cm, then what is the sum of the lengths (in cm) of the direct common tangent and the transverse common tangent?
An equilateral triangle ABC and a scalene triangle DBC are inscribed in a circle on same side of the arc. what is ∠BDC equal to?
What can be the maximum number of common tangent which can be drawn to two non-intersecting circles?
There are two identical circles of radius 10 cm each. If the length of the direct common tangent is 26 cm, then what is the length (in cm) of the transverse common tangent?
