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 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.

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.

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.