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.]

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 $\angle PTQ$, so OT is perpendicular to PQ, bisecting PQ at R. Therefore, PR is half of PQ.

Step 2: Applying Pythagorean Theorem in $\triangle$ OPR

In right-angled triangle $\triangle OPR$, 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 $5^2 = 25$ and $4^2 = 16$. Subtract 16 from 25 to find $OR^2$. 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 $\angle OPT = 90^\circ$. Also, we established that OT is perpendicular to PQ, so $\angle ORP = 90^\circ$. Triangles $\triangle OPR$ and $\triangle OPT$ 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 $\triangle OPR$ and $\triangle OPT$, 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 $\frac{20}{3}$ cm.

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