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Finding angles involving tangents and circles

In this lesson, we will look at finding angles in diagrams that involve tangents and circles.

Some of the theorems used are:
Tangent to Circle Theorem
Pythagorean Theorem
Two-Tangent Theorem

The following diagram shows the properties of the line segments and angles formed by the tangents from a point outside a circle. Scroll down the page for more examples and solutions on how to use the properties to solve for angles.

tangents to circle


In the following diagram, PA and PB are tangents to the circle. Find the value of:

a) ∠OAP
b) ∠AOB
c) ∠OBA
d) ∠ASB
e) the length of OP, given PB = 7 cm.

tangent problems

Solution :

How to solve for unknown values using the properties of a tangent line to a circle?
Tangent to Circle Theorem
A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.
How to solve for unknown values using the properties of tangent segments to a circle from a given point?
Tangent to a Circle from a Point
Tangent segments to a circle from a point are congruent. Tangent Lines to a Circle
Tangent lines touches the circle at only one point
Point of Tangency is where the tangent line touches the circle
Is AB a tangent line?
Find the missing angles. How to use the Properties of Tangents to find missing angles or sides?
Terms to know
A circle is a set of all points in a plane equidistant from a given point.
A radius is a segment whose endpoints are the center and a point on the circle.
A chord is a segment whose endpoints are on the circle.
A diameter is a chord that contains the center.
A secant is a line that a circle at two points.
A tangent is a line that intersects the circle at one point (point of tangency).
A common tangent is a line, ray or segment that is tangent to two coplanar circles.

Tangents to Circles
1. You are standing 14 feet from a water tower. The distance from you to the point of tangency on the tower is 28 feet. What is the radius of the water tower?
2. AB is a tangent to circle C at B. AD is a tangent to circle C at D. Find the value of x. How to find angles in diagrams involving tangents and circles?

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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