95 Tangents to Circles Objectives Identify segments and lines related to circles Use properties of a tangent to a circle Some definitions you need Circle set of all points in a plane that are ID: 419213
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Slide1
Geometry
9.5
Tangents to CirclesSlide2
Objectives
Identify segments and lines related to circles.
Use properties of a tangent to a circle
.Slide3
Some definitions you need
Circle
– set of all points in a plane that are
equidistant
from a given point called a
center of the circle. A circle with center P is called “circle P”, or P.The distance from the center to a point on the circle is called the radius of the circle. Two circles are congruent if they have the same radius.Slide4
Some definitions you need
The distance across the circle, through its center is the
diameter
of the circle. The diameter is twice the radius.
The terms
radius and diameter describe segments as well as measures.Slide5
Some definitions you need
A radius is a segment whose endpoints are the center of the circle and a point on the circle
.
QP, QR, and QS are radii of Q. All radii of a circle are congruent.Slide6
Some definitions you need
A
chord
is a segment whose endpoints are points on the circle. PS and PR are
chords
.A diameter is a chord that passes through the center of the circle. PR is a diameter.Slide7
Some definitions you need
A
secant
is a line that intersects a circle in two points.
Line
k is a secant.A
tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line
j is a tangent. Slide8
Identifying
Special Segments and Lines
Tell
whether the line
or segment
is best described as a chord, a secant, a tangent, a diameter, or a radius of C.ADCDEG
HBSlide9
More information you need--
In a plane, two circles can intersect in
two points, one point, or no points.
Coplanar circles that intersect in one point are called
tangent circles
. Coplanar circles that have a common center are called concentric.
2 points of intersection.Slide10
Tangent circles
A line or segment that is tangent to two coplanar circles is called a common tangent.
A common internal tangent intersects the segment that joins the centers of the two circles.
A
common external tangent does not intersect the segment that joins the center of the two circles.
Internally tangent
Externally tangentSlide11
Concentric circles
Circles that have a common center are called concentric circles.
Concentric circles
No points of intersectionSlide12
Identifying
common tangents
Tell whether the common tangents are internal or external. Slide13Slide14Slide15
Verifying
a Tangent to a Circle
Determine if EF
is tangent to
circle D.
Because 112 + 602 = 612
∆
DEF is a right triangle and DE is perpendicular to EF.
EF
is tangent to
circle D
.Slide16
Find Radius
(r + 8)
2
= r
2
+ 16
2
Pythagorean Thm.
Substitute values
c
2
= a
2
+ b
2
r
2
+ 16r + 64 = r
2
+ 256
Square of binomial
16r + 64 = 256
16r = 192
r = 12
Subtract r
2
from each side.
Subtract 64 from each side.
Divide.
The radius of the
circle
is 12 feet.Slide17
Using properties of tangents
AB is tangent to C at B.
AD is tangent
to
C at D.
Find the value of x.
x
2
+ 2Slide18
x
2
+ 2
11 = x
2
+ 2
Two tangent segments from the same point are
Substitute values
AB = AD
9 = x
2
Subtract 2 from each side.
3 = x
Find the square root of 9.
The value of x is 3 or -3.
AB is tangent to C at B.
AD is tangent
to
C at D.
Find the value of x.
x
2
+ 2Slide19
PracticeSlide20
X ²= 400+ 225
x
² = 625
X = 25Slide21
40² + 30
² = 2500
Hypotenuse = √2500 = 50
X
= 50-30 =
20
Hypotenuse = 21 + 8 = 29
29
² - 21 ² = 400
X
=
√400 =
20Slide22
AC = BC
40 = 3x + 4
36 = 3x
X = 9
Pythagorean Formula
BP² = AB
² + AP²
(8 + x)
² = x ² + 12²
x
² + 16x + 64 = x² + 144
16x = 80
X = 5Slide23
CD = ED = 4
BC = AB = 3
X = BC + CD
X
= 3+4 =
7
Trigonometric Formula
Cos 30 = 6 / x x = 6 / Cos 30 x = 6.9
30°Slide24
Pythagorean Formula
BP² = CP
² + CB²
X
² = 12 ² - 4²
x² = 144-16X = 11.3
PB = 6
PC = 6 + 4 = 10
PQ = 6
Pythagorean Formula
QC
² = PC ² -PQ ²
QC
²
= 100-36 = 64
QC = 8
Pythagorean Formula
CD
² = CQ ² - QD ²
CD
² = 64 – 40.96
CD = 4.8