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P is a point on the circumference of a circle. From P, perpendiculars PA and PB are dropped onto two mutually perpendicular diameters, meeting them at A and B. If AB = 36 cm, find the diameter of the circle.

1. 72 cm
2. 36 cm
3. 18 cm
4. 24 cm

Correct answer: 72 cm

Solution

Let the two perpendicular diameters be the x- and y-axes with centre O at the origin. Let P = (x, y) lie on the circle, so x² + y² = r². The foot of perpendicular from P to the x-axis is A = (x, 0); to the y-axis is B = (0, y). Then OAPB is a rectangle, and its diagonal AB equals OP = sqrt(x² + y²) = r. So AB = r = 36 cm, hence diameter = 2r = 72 cm.

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