Question

Let PQ and RS be tangents at the extremities of diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals_______

Answer 1

The length of 2r, representing the diameter of a circle with radius r, where tangents meet the ends of this diameter, is equivalent to the lengths of the tangents (PQ and RS) from their points of tangency to their intersection with an extended diameter on the circumference (at point X).

Step-by-step explanation:

The question relates to the properties of tangents and diameters in circles. Given that PQ and RS are tangents at the extremities of diameter PR of a circle of radius r, and PS and RQ intersect at point X on the circumference, the problem suggests we use the circle's properties to find the length of 2r, that is, the diameter of the circle. According to the question's conditions, point X must lie on the circle precisely in the middle of PQ and RS, as the angles at which the tangents meet the diameter will be right angles.

This implies that PX and RX are both radii of the circle and, therefore, are equal in length to r. Since PX and RX form part of the diameters PQ and RS, respectively, and X is the midpoint of both, it follows that the length of both PQ and RS is twice the radius. Subsequently, 2r represents the diameter of the circle, which is equal to the lengths of PQ and RS.