PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. [Figure: circle centre O radius 5 cm, chord PQ = 8 cm, tangents at P and Q meeting at external point T; OT meets PQ at R.]
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Step-by-Step Solution
Step 1: Bisecting the chord and forming right triangle
Given that PQ is a chord of length 8 cm, and we know that the line segment from the center to the chord which is perpendicular to the chord, bisects the chord. OT is the angle bisector of ∠PTQ, so OT is perpendicular to PQ, bisecting PQ at R. Therefore, PR is half of PQ.
Step 2: Applying Pythagorean Theorem in △ OPR
In right-angled triangle △OPR, where OP is the radius, OR is the distance from the center to the chord, and PR is half the chord length. We can use the Pythagorean theorem to find the length of OR. We are given OP=5 cm and we calculated PR=4 cm.
Step 3: Calculating OR
Substitute the values of OP and PR into the Pythagorean theorem equation. Calculate 52=25 and 42=16. Subtract 16 from 25 to find OR2. Taking the square root of 9 gives us the length of OR.
Step 4: Relating TP, OP, and TR using similar triangles
The radius OP is perpendicular to the tangent TP at the point of contact P, so ∠OPT=90∘. Also, we established that OT is perpendicular to PQ, so ∠ORP=90∘. Triangles △OPR and △OPT are similar by AA similarity criterion (Angle-Angle similarity). This means their corresponding sides are proportional.
Step 5: Using proportionality of similar triangles to find TP
From the similarity of △OPR and △OPT, we can set up a proportion of their corresponding sides. We want to find TP, and we know RP, OP, and OR. Rearranging the proportion allows us to solve for TP. Substitute the known values RP=4 cm, OP=5 cm, and OR=3 cm.
Step 6: Calculating TP
By substituting the values RP=4 cm, OP=5 cm, and OR=3 cm into the proportionality equation, we can calculate the length of TP. The length of TP is found to be 320 cm.