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Mathematics · Circles · NCERT Examples

Example 3

A complete, independently verified solution for NCERT Example 3.

Mathematics · Chapter 10 · NCERT Examples

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NCERT Example

Example 3

chord · tangent length

PQ is a chord of length 8 cm in a circle of radius 5 cm. The tangents at P and Q meet at T. Find TP.

Chord PQ of length 8 centimetres in a circle of radius 5 centimetres, with tangents meeting at T.
Figure 10-10 from the supplied NCERT chapter.

Solution

Board-exam working:

Join OT and let it meet PQ at R.

Since TP = TQ, triangle TPQ is isosceles. OT bisects ∠PTQ and is perpendicular to PQ.

PR=RQ=82=4 cmPR=RQ=\frac82=4\text{ cm}

In right triangle OPR:

OR=OP2−PR2OR=\sqrt{OP^2-PR^2}
=52−42=3 cm=\sqrt{5^2-4^2}=3\text{ cm}

Triangles TRP and PRO are similar by AA similarity.

TPPO=PRRO\frac{TP}{PO}=\frac{PR}{RO}
TP5=43\frac{TP}{5}=\frac43
TP=203 cmTP=\frac{20}{3}\text{ cm}

Final answer

TP=203 cmTP=\frac{20}{3}\text{ cm}