PQ is a chord of length 9cm of a circle of radius 6 cm. The tangents to the circle at points P and Q intersect each other at an external point T. Find the length of tangent TP.
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Step-by-Step Solution
Step 1: Understand the Geometry
We are given a circle with center O and radius OP=OQ=6 cm. A chord PQ has a length of 9 cm. Tangents from an external point T touch the circle at P and Q. We need to find the length of the tangent TP.
Step 2: Identify Key Properties and Triangles
From the properties of tangents, we know that tangents drawn from an external point to a circle are equal in length, so TP=TQ. Also, the radius is perpendicular to the tangent at the point of contact, so ∠OPT=90∘. This means △OPT is a right-angled triangle.
Step 3: Find the area of triangle OPQ using Heron's formula
First, we find the semi-perimeter s of △OPQ. Then, we use Heron's formula to calculate the area of △OPQ. This area will be useful in finding the altitude from O to PQ.
Step 4: Find the altitude from O to PQ
Let OM be the altitude from O to PQ. We can also express the area of △OPQ as 21×base×height. Using the calculated area, we can find the length of OM.
Step 5: Find PM using Pythagorean theorem in triangle OMP
The perpendicular from the center to a chord bisects the chord, so PM=21PQ=29 cm. We can verify this using the Pythagorean theorem in △OMP. This step confirms our calculations for OM and PM.
Step 6: Use similar triangles to find TP
Consider △OMP and △OPT. Both are right-angled triangles. ∠OMP=90∘ and ∠OPT=90∘. Also, ∠POM is common to both triangles. Therefore, △OMP∼△OPT by AA similarity. Using the ratio of corresponding sides, we can find the length of PT.