10'1: Tangents to Circles

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10.1 Tangents to Circles

• Objective To identify segments and lines related
  to circles and use properties of tangents to
  circles.

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What is a circle?

• Circle the set of all points in a plane that
  are equidistant from a given point.
• Center the given point in which the set of
  points is equidistant.
• Radius the distance from the center to a point
  on the circle.
• Diameter the distance across the circle and
  through the diameter.
• The radius and diameter also represent segments
  on the circle.

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For the visual learner
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More parts of a circle

• Chord a segment whose endpoints are points on
  the circle.
• A diameter is also a cord but it is special
  because it passes through the center.
• Secant a line that intersects a circle in two
  points.
• Tangent a line that intersects a circle in
  exactly on point.

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Now for a lovely diagram.
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Example

• Identify the parts of the following circle.

BE BH BC DG AF
Chord Diameter Radius Tangent Secant
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Types of circles and other info.

• Tangent circles coplanar circles that intersect
  in one point.
• Concentric circles coplanar circles which have
  no points of intersection
• Congruent circles Two circles are congruent if
  they have the same radius.

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Types of circles.
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One more type of circle.
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Common Tangents

• Common tangent a line or segment that is
  tangent to two circles.

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Interiors and Exteriors

• Interior of a circle the points that are inside
  the circle
• Exterior of a circle the points that are
  outside the circle.

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Describing circles.

• Circle A Center (3,2) Radius 2
• Circle B Center (7,2) Radius 2
• These two circles are congruent!
• They intersect at (6,2) and have common tangent
  lines of y4, y0, and x5.

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Theorems

• If a line is tangent to a circle, then the radius
  drawn to the point of tangency is perpendicular
  to the tangent.
• Note that the converse is also true!

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Theorems

• If two segments drawn from the same exterior
  point of a circle and they are both tangents to
  the circle, then the segments are congruent.

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Examples

• Is CE tangent to circle D?

If we have a right triangle, then CE would be
perpendicular to ED and that would be a tangent
by definition. How can we find out whether or not
we have a right triangle? Answ Use the Converse
of the Pythagorean Theorem.
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Examples

• You are standing 14 feet from a water tower. The
  distance from you to a point of tangency on the
  tower is 28 feet. What is the radius of the
  water tower?

Use the Pythagorean Theorem to solve for r!