definition
A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point.
DISCUS THROW
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:


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)
4tans
2 external tangents (blue)
2 internal tangents (black)
3 Common Tangents
(2 externally tangent circles)
3tans
2 external tangents (blue)
1 internal tangent (black)
2 Common Tangents
(2 overlapping circles) 2tans
2 external tangents (blue)
0 internal tangents
1 Common Tangent
(2 internally tangent circles)1trans1
1 external tangent (blue)
0 internal tangents

0 Common Tangents
(2 concentric circles)
Concentric circles are circles
with the same center.
0trans2
0 external tangents
0 internal tangents
(one circle floating inside
the other, without touching)

0trans
0 external tangents
0 internal tangents

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theorem
If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
 
thm1
tanm1

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theorem
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.)
 
thm2
tanm2
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.
 tanm3

 


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