Two circles have the same centre and their radii are 15cm and 17cm. A tangent to the inner circle at p cuts the outer circle at Q. Calculate pQ
Since both circles have the same center, line pQ passes through the center of the circles. Therefore, line pQ is perpendicular to the tangent line at point p. Let pQ = x.
According to the properties of tangents, the tangent line is perpendicular to the radius drawn from the center of the circle to the point of tangency. Therefore, the tangent line at point p is perpendicular to line OQ, where O is the center of the circles.
In the right triangle OPQ, where O is the center of the circles, OP is the radius of the smaller circle, and OQ is the radius of the larger circle. By the Pythagorean theorem, we have:
OP^2 + PQ^2 = OQ^2
Substituting the given values, we get:
15^2 + x^2 = 17^2
225 + x^2 = 289
Simplifying the equation, we have:
x^2 = 289 - 225
x^2 = 64
Taking the square root of both sides, we have:
x = √(64)
x = 8
Therefore, pQ = 8 cm.