Geometry

April 19, 2023

Goals

Know properties of circles. Identify special lines in a circle. Solve problems with special lines.

April 19, 2023

Circle: Set of points on a plane equidistant from a point (center).

C

This is circle C, or

C

R

CR is a radius.

A

B

AB is a diameter.

The diameter is twice the radius.

April 19, 2023

Terminology

One radius Two radii radii = ray-dee-eye

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All Radii in a circle are congruent

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Interior/Exterior

AA is in the interior of the circle.

B

B is in the exterior of the circle.

C

C is on the circle.

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Congruent Circles

Radii are congruent.

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April 19, 2023

Chord

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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Secant

A secant is a line that intersects a circle at two points.

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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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Tangent Circles

Intersect at exactly one point.

These circles are externally tangent.

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Tangent Circles

Intersect at exactly one point.

These circles are internally tangent.

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Can circles intersect at two points?

YES!

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Concentric Circles

Have the same center, different radius.

April 19, 2023

Concentric Circles

Have the same center, different radius.

April 19, 2023

Concentric Circles

Have the same center, different radius.

April 19, 2023

Concentric Circles

Have the same center, different radius.

April 19, 2023

Concentric Circles

Have the same center, different radius.

April 19, 2023

Concentric Circles

Have the same center, different radius.

April 19, 2023

Concentric Circles

Have the same center, different radius.

April 19, 2023

Concentric Circles

Have the same center, different radius.

April 19, 2023

Common External Tangents

This is a common external tangent.

And this is a common external tangent.

April 19, 2023

Common External Tangents in a real application…

April 19, 2023

Common Internal Tangents

This is a common internal tangent.

And this is a common internal tangent.

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April 19, 2023

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.

April 19, 2023

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)

April 19, 2023

Example 1

R A

T

5

12

13

YES

TA = 13RAT is a right triangle.

52 + 122 = 132

25 + 144 = 169

169 = 169

Is RA tangent to T?

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FOILFind (x + 3)2

(x + 3)(x + 3)

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FOILFind (x + 3)2

(x + 3)(x + 3)

x2

April 19, 2023

FOILFind (x + 3)2

(x + 3)(x + 3)

x2

3x

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FOILFind (x + 3)2

(x + 3)(x + 3)

x2 + 3x

3x

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FOILFind (x + 3)2

(x + 3)(x + 3)

9

x2 + 3x + 3x

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FOILFind (x + 3)2

(x + 3)(x + 3)

x2 + 3x + 3x + 9

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FOIL(x + 3)2 = x2 + 6x + 9

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Expand (x + 9)2

(x + 9)(x + 9) F: x2

O: 9x I: 9xL: 81 (x + 9)2 = x2 + 18x + 81

April 19, 2023

Example 2BC is tangent to circle A at B. Find r.

A

B C

r16

24

D

DC = 16

rAC = ?AC = r + 16

r2 + 242 = (r + 16)2

April 19, 2023

r2 + 242 = (r + 16)2

r2 + 576 = (r + 16)(r + 16)

r2 + 576 = r2 + 16r + 16r + 256

576 = 32r + 256

320 = 32r

r = 10

Solve the equation.

r2 + 242 = (r + 16)2

April 19, 2023

A

B C

1016

24

D

AC = 26

10

Here’s where the situation is now.

Check:

102 + 242 = 262

100 + 576 = 676

676 = 676

26

r = 10

April 19, 2023

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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Example 3

HE and HA are tangent to the circle. Solve for x.

H

A

E

12x + 15

9x + 45

April 19, 2023

Solution

12x + 15 = 9x + 45

3x + 15 = 45

3x = 30

x = 10H

A

E

12x + 15

9x + 459(10) + 45

90 + 45 = 135

12(10) + 15

120 + 15 = 135

April 19, 2023

Try This:

The circle is tangent to each side of ABC. Find the perimeter of ABC.

A

BC

7

2

5

2

57

9 7

12

7 + 12 + 9 = 28

April 19, 2023

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?

April 19, 2023

Practice Problem 1

MD and ME are tangent to the circle. Solve for x.

D

E

M

4x 12

2x + 12

4x – 12 = 2x + 12

2x – 12 = 12

2x = 24

x = 12

April 19, 2023

Practice Problem 2

x2 + 42 = (4 + 12)2

x2 + 16 = 256

x2 = 240

x = 415 15.5

R

T

4

12

x

Solve for x.

April 19, 2023

Practice Problem 3

x2 + 82 = (x + 6)2

x2 + 64 = x2 + 12x + 36

64 = 12x + 36

28 = 12x

x = 2.333…

R

T6

8

x

Solve for x.

x

April 19, 2023

Practice Problems