Tangent pipes to a circular tank: equal lengths and a cyclic quadrilateral
A circular water tank of radius 8 m stands in a level rectangular field. Its centre is O. A straight service pipe PQ touches the tank at P, and the valve point Q is 17 m from O. From the same valve Q, a second pipe QR touches the tank at R. A worker claims that because the two pipes come from the same valve, PQ and QR must be equal, but the site supervisor wants the claim proved rather than assumed. The supervisor also needs the area and perimeter of the quadrilateral OPRQ enclosed by the two radii and the two pipes. Finally, the supervisor wants to know whether O, P, Q and R lie on one circle; if they do, she wants the radius of that circle. All pipe lengths are in metres and the ground is level.
1. Prove that PQ=QR and find the common length.
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Since PQ is tangent at P, OP is perpendicular to PQ, so angle OPQ = 90°. Similarly, OR is perpendicular to QR, so angle ORQ = 90°. In triangles OPQ and ORQ: OP = OR = 8 m (radii), OQ is common, and both triangles are right-angled at P and R respectively. By the RHS congruence criterion, triangle OPQ is congruent to triangle ORQ. Hence PQ = QR. In right triangle OPQ, PQ^2 = OQ^2 - OP^2 = 17^2 - 8^2 = 289 - 64 = 225, so PQ = 15 m. Therefore QR = 15 m as well.
2. Find the area and perimeter of quadrilateral OPRQ.
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The quadrilateral is made of two congruent right triangles, OPQ and ORQ. Area of triangle OPQ = (1/2)(OP)(PQ) = (1/2)(8)(15) = 60 m^2. The two triangles have equal area, so area of OPRQ = 2 x 60 = 120 m^2. Its perimeter is OP + PQ + QR + RO = 8 + 15 + 15 + 8 = 46 m.
3. Show that O, P, Q and R lie on one circle, and find the radius of that circle.
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We have angle OPQ = 90° and angle ORQ = 90°. These are opposite angles of quadrilateral OPRQ, and their sum is 90° + 90° = 180°. A quadrilateral is cyclic if a pair of opposite angles is supplementary, so O, P, Q and R lie on one circle. Moreover, angle OPQ = 90° means that OQ subtends a right angle at P, so OQ is a diameter of the circumcircle. The radius is therefore OQ/2 = 17/2 = 8.5 m. The circle has diameter OQ, not radius 8 m; the tank radius is a separate length.