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A tangent line to a circle plays an important role in our investigation of circles,
even though it never enters the interior of the circle.
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A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. |
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The discus throw is a track and field event in which an athlete attempts to throw a heavy disc (called a discus) further than his or her competitors. The athlete spins counterclockwise around one and a half times through a circle, then releases the throw. When released, the discus travels on a path that is tangent to the circular spin orbit. |
Common tangents are lines, rays or segments that are tangent
to more than one circle at the same time.
4 Common Tangents
(2 completely separate circles)
2 external tangents (blue)
2 internal tangents (black)
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3 Common Tangents
(2 externally tangent circles)
2 external tangents (blue)
1 internal tangent (black)
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2 Common Tangents
(2 overlapping circles)
2 external tangents (blue)
0 internal tangents
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1 Common Tangent
(2 internally tangent circles)
1 external tangent (blue)
0 internal tangents
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0 Common Tangents
(2 concentric circles)
Concentric circles are circles
with the same center.
0 external tangents
0 internal tangents
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(one circle floating inside
the other, without touching)
0 external tangents
0 internal tangents
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If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency. |
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Example:
Given a tangent with a radius
drawn to the point of tangency.
Find x.
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Solution:
• In this diagram, the radius is perpendicular to the tangent at the point of tangency, forming the right triangle shown.
Since OB = 6, the missing portion to complete OA is also 6 (a radius length), making OA = 10.
Using Pythagorean Theorem, x = 8.
c2 = a2 + b2
102 = 62 + x2
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Tangent segments to a circle from the same external point are congruent.
(You may see this theorem referred to as the "hat" theorem as the circle appears to be wearing a hat.) |
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Example:
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Solution:
• In this diagram, AB = AC.
2x + 15 = 45
2x = 30
x = 15
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PROOF:
With the help of the previous theorem, this theorem can be easily proven. The triangles shown at the right are congruent by Hypotenuse Leg for Right Triangles. The radii of a circle are congruent (the legs), and the triangles share a side (the hypotenuses). The triangles have right angles at A and C since a radius drawn to the point of tangency is perpendicular to the tangent. By CPCTC, the tangents are congruent. |
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