9.6 Solving Right Triangles

1
What is Unit 3 about?
We will learn to use arcs, angles, and segments
in circles to solve real life problems. We will
learn how to find the measure of angles related
to a circle. We will find circumference and
area of figures with circles. We will also find
the surface and volume of spheres.
2
Crop Circles
Whether you think crop circles are made by little
green men from space or by sneaky earthling
geeks, you've got to admit that they are pretty
dang cool...  And whoever is making them knows a
ton of geometry!
http//www.coolmath.com/lesson-geometry-of-crop-ci
rcles-1.html
3
Properties of Tangents
Friday, October 04, 2013

• Essential Question
• How do we identify segments and lines related to
  circles and how do we use properties of a tangent
  to a circle?

Lesson 6.1
M2 Unit 3 Day 1
4
Warm Ups
1. What measure is needed to find the
circumference or area of a circle?
2. Find the radius of a circle with diameter 8
centimeters.
3. A right triangle has legs with lengths 5
inches and12 inches. Find the length of the
hypotenuse.
5. Solve (x 18)2 x2 242.
4. Solve 6x 15 33.
5
(No Transcript)
6
Circle
The set of all points in a plane that are
equidistant from a given point, called the center.
E
F
Tangent
P
D
A
Secant
C
B
Name of the circle ? P
7
Definition

• Radius a segment from the center of the circle
  to any point on the circle.

• Diameter a chord that contains the center of
  the circle.

C
A
B
8
Definition

• Chord a segment whose endpoints are points on
  the circle.

9
Definition

• Secant a line that intersects a circle in two
  points.

10
Definition

• Tangent a line in the plane of a circle that
  intersects the circle in exactly one point.

P
Point of tangency
O
11
A little extra information

• The word tangent comes from the Latin word
  meaning to touch
• The word secant comes from the Latin word meaning
  to cut.

12
EXAMPLE 1

• 1. Tell whether the line or segment is best
  described as a chord, a secant, a tangent, a
  diameter, or a radius.

tangent
diameter
chord
radius
13
EXAMPLE 2
Identify special segments and lines
SOLUTION
14
EXAMPLE 3
Find lengths in circles in a coordinate plane
SOLUTION
15
GUIDED PRACTICE
EXAMPLE 4
SOLUTION
16
Definitions

• Common tangent a line or segment that is
  tangent to two coplanar circles
• Common internal tangent intersects the segment
  that joins the centers of the two circles
• Common external tangent does not intersect the
  segment that joins the centers of the two circles

17
EXAMPLE 5

• 5. Tell whether the common tangents are internal
  or external.

a.
b.
common internal tangents
common external tangents
18
Draw common tangents
EXAMPLE 6
6. Tell how many common tangents the circles have
and draw them.
SOLUTION
19
Perpendicular Tangent Theorem 6.1

• In a plane, if a line is tangent to a circle,
  then it is perpendicular to the radius drawn to
  the point of tangency.

20
Tangent Theorems
Create right triangles for problem solving.
21
EXAMPLE 8
Find the radius of a circle
AC2 BC2 AB2
Pythagorean Theorem
Substitute.
(r 50)2 r2 802
Write the binomial twice.
(r 50)(r 50) r2 802
Multiply.
r2 50r 50r 2500 r2 6400
Combine Like Terms.
r2 100r 2500 r2 6400
100r 3900
Subtract from each side.
r 39 ft .
Divide each side by 100.
22
EXAMPLE 8
QT2 QS2 ST2
Pythagorean Theorem
(r 18)2 r2 242
Substitute.
r2 36r 324 r2 576
Multiply.
36r 252
Subtract from each side.
r 7
Divide each side by 36.
23
Perpendicular Tangent Converse

• In a plane, if a line is perpendicular to a
  radius of a circle at its endpoint on the circle,
  then the line is tangent to the circle.

24
GUIDED PRACTICE
EXAMPLE 10
25
Verify a tangent to a circle
EXAMPLE 7
26
Congruent Tangent Segments Theorem 6.2

• If two segments from the same exterior point are
  tangent to a circle, then they are congruent.

27
11.
SOLUTION
RS RT
28 3x 4
Substitute.
8 x
Solve for x.
28
12.
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Homework Page 187 18 24 all Page 188 1
10.