PQ is a chord of length 8 cm of a circle of radius 5 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.

Step 1: Draw the diagram and identify key properties

First, let's visualize the problem. We have a circle with center O and radius 5 cm. A chord PQ of length 8 cm is drawn. Tangents from points P and Q meet at an external point T. We need to find the length of the tangent TP. We know that the radius is perpendicular to the tangent at the point of contact. Also, the line segment OT bisects the chord PQ and is perpendicular to it.

Step 2: Find the length of PR using the chord property

Since the line segment OT bisects the chord PQ, it divides PQ into two equal halves. Let R be the point where OT intersects PQ. Therefore, PR is half the length of PQ.

Step 3: Calculate OR using the Pythagorean theorem in $\triangle$ OPR

In the right-angled triangle $\triangle OPR$, we know the hypotenuse OP (radius) and the side PR. We can use the Pythagorean theorem to find the length of OR.

Step 4: Prove similarity of triangles $\triangle$ OPR and $\triangle$ PRT

We know that $\angle OPT = 90^\circ$ (radius is perpendicular to tangent). In $\triangle OPR$, $\angle ORP = 90^\circ$. In $\triangle PRT$, $\angle PRT = 90^\circ$. By comparing angles, we can show that $\triangle OPR$ is similar to $\triangle PRT$.

Step 5: Find TP using similar triangles

Since $\triangle OPR \sim \triangle PRT$, the ratio of their corresponding sides will be equal. We can set up a proportion to find the length of TP.

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