Given : Two circles, both with centre M. PT is a chord of the larger circle cutting the smaller circle at Q and S. MR is perpendicular to PT. MP =17 units, MS =10 units and QS =12 units.

Question

calculate :
a) the length of MR
b) the length of PT

Answer 1

To solve this problem, we can use the properties of circles and right triangles.

a) To find the length of MR, we can use the Pythagorean Theorem in triangle MRS. Since MR is perpendicular to PT, triangle MRS is a right triangle.

Using the Pythagorean Theorem:
MR^2 + RS^2 = MS^2
MR^2 + 12^2 = 10^2
MR^2 + 144 = 100
MR^2 = 100 - 144
MR^2 = -44

Since the square of a real number cannot be negative, there must be an error in the given information or setup of the problem. It is not possible to find the length of MR using the given information.

b) To find the length of PT, we can use the Pythagorean Theorem in triangle PTS. Since PT is a chord of the larger circle, it passes through the center M of both circles. Therefore, triangle PTS is also a right triangle.

Using the Pythagorean Theorem:
PT^2 = PS^2 + ST^2
PT^2 = 12^2 + 17^2
PT^2 = 144 + 289
PT^2 = 433

Taking the square root of both sides:
PT = √433 ≈ 20.8 units

Therefore, the length of PT is approximately 20.8 units.