Math 20P
Polygon Angles
Inscribed - Inscribed Angle
Inscribed - Central Angle
Perpendicular Chord Bisector
Tangent to a Circle
Prerequisite Skills

Tangent to A Circle Properties (3)

Complete the Learning Strategy first to see the pattern, then read Circle Property to place what you understand in a mathematical context..

Circle Property

Learning Strategy

 

1. A 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 90o).
    • 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.

Download the newest java version if the applet fails to launch.

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

 

Interactive Activity

  • Change the size of the inscribed triangle by moving the vertices.
  • <A will always equal <B.
  • Hold the SHIFT key when you drag on the circumference of the circle to change the size of the circle.

 

Tangent to a Circle
 
 
Comments to:  Jim Reed
Started September, 1998. Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007