Math Problem Statement

Given a circle (O) with diameter AB, on one side of the line AB, two tangents Ax and By are drawn, both of which are perpendicular to the line AB. Let M be a point on the line Ax, which is not on the circle, and the tangent from point M touches the circle at N. a) Calculate MON. b) Prove that MN = AM + BN. c) Prove that AM . BN = R^2.

Solution

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Problem 24:

Given a circle OO with diameter ABAB, on one side of the line ABAB, the tangents AxAx and ByBy are drawn, both of which are perpendicular to the line ABAB. Let MM be a point on the line AxAx, which is not on the circle, and the tangent from point MM touches the circle at NN.

a) Calculate MONMON.

b) Prove that MN=AM+BNMN = AM + BN.

c) Prove that AMBN=R2AM \cdot BN = R^2, where RR is the radius of the circle.

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Math Problem Analysis

Mathematical Concepts

Circle geometry
Tangents
Perpendicular lines
Length of tangents from an external point to a circle

Formulas

Tangential length theorem
Power of a point theorem

Theorems

Power of a point theorem
Segment addition postulate

Suitable Grade Level

Grades 10-12