SS2 Mathematics Third Term: Circle Theorem: Tangent Properties of Circle

Introduction

A Tangent of a Circle has two defining properties:

Since a tangent only touches the circle at exactly one and only one point, that point must be perpendicular to a radius.

To test out the interconnected relationship of these two defining traits of a tangent, try the interactive applet.

The point where the tangent and the circle intersect is called the point of tangency.

Tangent to a circle is perpendicular to the radius at the point of tangency.

This is a very useful property when the radius that connects to the point of tangency is part of a right angle, because the trigonometry and the Pythagorean Theorem apply to right triangles.

Vocabulary:

A tangent intersects a circle at one point.
- C and D are the points of tangency to circle O
- AC and AD are tangent to circle O.
Perpendicular means at right angles (meet at 90^o).
- OC and OD are radii of the circle O.
- OC is perpendicular to AC.
- OD is perpendicular to AD.

Interactive Activity

Adjust the positions of the tangents (see diagram on the left) by dragging the point A.
Adjust the radius position by dragging points C and D.
<C will always be perpendicular to tangent AC and <D will always be perpendicular to tangent AD.
Move point B to overlap radius OC or OD. Radius BO will be perpendicular to the corresponding tangent line.
Hold the SHIFT key when you drag on the circumference of the circle to change the size of the circle.

2. The tangent segments to a circle from an external point are equal.

Tangent Segement Proof

Interactive Activity

Adjust the positions of the tangents by dragging the external point A.
Adjust the radius position by dragging points C and D.
The length of segment AC will always equal the length of segment AD.
Hold the SHIFT key when you drag on the circumference of the circle to change the size of the circle.

3. The angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord.

Point A is the point of tangency (point where the tangent line touches the circle) of line AX.
Chord AC is a segment with endpoints on the circle.
<XAC is the angle between tangent AX and chord AC.
<ABC is the angle opposite chord AC.
The animation in frame 3/3 shows that the size of <ABC does not change when you move the vertex (point B).
Eventually <ABC will lay on top of <XAC. This shows the two angles must be congruent.
The interactive on the left allows you switch the positions of <ABC and <XAC. In each case <ABC can be moved on top of <XAC. This shows the two angles must be congruent. If you can remember what you see, you will likely remember: The angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord.

Tangent Chord