1. Geometry
Circles Circles

2. Goals Know properties of circles. Identify special lines in a circle.
Solve problems with special lines.
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3. Circle: Set of points on a plane equidistant from a point (center).B
This is circle C, or C
CR is a radius.
AB is a diameter. R A
The diameter is twice the radius.
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4. Terminology
One radius
Two radii
radii = ray-dee-eye
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5. All Radii in a circle are congruent
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6. Interior/Exterior A is in the interior of the circle. A
C is on the circle. C
B is in the exterior of the circle. B
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7. Congruent Circles
Radii are congruent.
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8. Lines in
a circle.
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9. Chord A diameter is a chord that passes through the center.
A chord is a segment between two points on a circle.
A diameter is a chord that passes through the center.
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10. Secant A secant is a line that intersects a circle at two points.
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11. Tangent
A tangent is a line that intersects a circle at only one point.
It is called the point of tangency.
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12. Tangent Circles Intersect at exactly one point.
These circles are externally tangent.
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13. Tangent Circles Intersect at exactly one point.
These circles are internally tangent.
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14. Can circles intersect at two points?
YES!
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15. Concentric Circles
Have the same center, different radius.
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16. Concentric Circles
Have the same center, different radius.
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17. Concentric Circles
Have the same center, different radius.
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18. Concentric Circles
Have the same center, different radius.
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19. Concentric Circles
Have the same center, different radius.
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20. Concentric Circles
Have the same center, different radius.
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21. Concentric Circles
Have the same center, different radius.
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22. Concentric Circles
Have the same center, different radius.
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23. Common External Tangents
And this is a common external tangent.
This is a common external tangent.
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24. Common External Tangents in a real application…
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25. Common Internal Tangents
And this is a common internal tangent.
This is a common internal tangent.
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26. Tangent
Theorems
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27. Theorem 12.1 (w/o proof)
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
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28. Theorem 12.2 (w/o proof)
If a line drawn to a circle is perpendicular to a radius, then the line is a tangent to the circle.
(The converse of 10.1)
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29. Example 1
Is RA tangent to T?
YES R 12 A 5
= 132
= 169
169 = 169 13 T
TA = 13
RAT is a right triangle.
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30. FOIL
Find (x + 3)2
(x + 3)(x + 3)
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31. FOIL
Find (x + 3)2
(x + 3)(x + 3) x2
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32. FOIL
Find (x + 3)2
(x + 3)(x + 3) 3x x2
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33. FOIL
Find (x + 3)2
(x + 3)(x + 3) 3x
x2 + 3x
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34. FOIL
Find (x + 3)2
(x + 3)(x + 3) 9
x2 + 3x + 3x
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35. FOIL
Find (x + 3)2
(x + 3)(x + 3)
x2 + 3x + 3x + 9
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36. FOIL
(x + 3)2 = x2 + 6x + 9
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37. Expand (x + 9)2 (x + 9)(x + 9) F: x2 O: 9x I: 9x L: 81
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38. Example 2 r2 + 242 = (r + 16)2 BC is tangent to circle A at B. Find r.
DC = 16
AC = r + 16
AC = ? r
r = (r + 16)2
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39. Solve the equation. r2 + 242 = (r + 16)2 r2 + 242 = (r + 16)2
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40. Here’s where the situation is now.26 A 10 D 16 10 C B 24
Check:
= 262
= 676
676 = 676
AC = 26
r = 10
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41. Theorem 12.3
If two segments from the same exterior point are tangent to a circle, then the segments are congruent.
Theorem Demo
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42. Example 3 HE and HA are tangent to the circle. Solve for x. A 12x + 15
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43. Solution 12x + 15 = 9x + 45 3x + 15 = 45 3x = 30 x = 10 12(10) + 15
= 135
12x + 15 = 9x + 45
3x + 15 = 45
3x = 30
x = 10 A
12x + 15 H
9x + 45
9(10) + 45
= 135 E
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44. Try This:
The circle is tangent to each side of ABC. Find the perimeter of ABC.
= 28 A 2 2 9 7 7 5 C B 7 5
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45. Can you… Identify a radius, diameter? Recognize a tangent or secant?
Define Concentric circles? Internally tangent circles? Externally tangent?
Tell the difference between internal and external tangents?
Solve problems using tangent properties?
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46. Practice Problem 1 MD and ME are tangent to the circle. Solve for x.
4x – 12 = 2x + 12
2x – 12 = 12
2x = 24
x = 12
4x  12 M
2x + 12 E
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47. Practice Problem 2 x2 + 42 = (4 + 12)2 x2 + 16 = 256 x2 = 240
Solve for x.
x = (4 + 12)2
x = 256
x2 = 240
x = 415  15.5
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48. Practice Problem 3 x2 + 82 = (x + 6)2 x2 + 64 = x2 + 12x + 36
Solve for x.
x = (x + 6)2
x = x2 + 12x + 36
64 = 12x + 36
28 = 12x
x = 2.333…
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49. Practice Problems
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