in the following fig two tangents PQ and PR are drawn to a circle with centre O from an external point P.prove that QPR=2OQR

To prove that angle QPR is twice angle OQR, we can use the properties of tangents and angles in circles. Here's an explanation on how to prove it step by step:

Step 1: Draw the figure
Draw a circle with center O. Draw two tangents PQ and PR from an external point P to the circle.

Step 2: Identify the angles
Label the points where the tangents touch the circle as point Q and point R, as shown in the figure. Also, label the point where the tangents intersect outside the circle as point P.

Step 3: Observe the angles
Notice that angle QPR is an exterior angle to triangle OQR, while angle OQR is an interior angle of triangle OQR.

Step 4: Recall the properties of exterior and interior angles
Remember that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior opposite angles. In this case, angle QPR is equal to angle OQR + angle ORQ.

Step 5: Apply the property to the given figure
Apply the property mentioned in step 4 to the figure. We have angle QPR = angle OQR + angle ORQ.

Step 6: Observe the angles again
Notice that angle ORQ is equal to angle OQR, since they are angles in the same triangle and opposite sides of the triangle are congruent.

Step 7: Substitute the equal angles in the equation
Replace angle ORQ with angle OQR in the equation from step 5. We get angle QPR = angle OQR + angle OQR, which simplifies to angle QPR = 2 * angle OQR.

Step 8: Conclusion
We have proved that angle QPR is twice the angle OQR, as required.

By following these steps, you can prove that QPR = 2OQR using the properties of tangents and angles in circles.