Transcript Tangent Lines LESSON 12-1 Additional Examples BA is tangent to . C at point A.
Tangent Lines LESSON 12-1 Additional Examples
BA
is tangent to
C
at point
A
. Find the value of
x
.
Because
BA
is tangent to
C
,
A
must be a right angle. Use the Triangle Angle-Sum Theorem to find
x
.
m A
+
m B
+
m C
= 180 90 + 22 +
x
= 180 Triangle Angle-Sum Theorem Substitute.
112 +
x
= 180
x
= 68 Simplify.
Solve.
Quick Check HELP GEOMETRY
Tangent Lines LESSON 12-1 Additional Examples
A belt fits tightly around two circular pulleys, as shown below. Find the distance between the centers of the pulleys. Round your answer to the nearest tenth.
HELP
Draw
OP
. Then draw
OD
parallel to
ZW
form rectangle
ODWZ
, as shown below. to Because
OZ
is a radius of
O
,
OZ
= 3 cm.
Because opposite sides of a rectangle have the same measure,
DW
= 3 cm and
OD
= 15 cm.
GEOMETRY
Tangent Lines LESSON 12-1 Additional Examples (continued) Quick Check
Because
ODP
is the supplement of a right angle,
ODP
is also a right angle, and
OPD
is a right triangle. Because the radius of
P
is 7 cm,
PD
= 7 – 3 = 4 cm.
OD
2 +
PD
2 =
OP
2 Pythagorean Theorem 15 2 + 4 2 =
OP
2 241 =
OP
2 Substitute.
Simplify.
OP
Use a calculator to find the square root.
The distance between the centers of the pulleys is about 15.5 cm.
HELP GEOMETRY
Tangent Lines LESSON 12-1 Additional Examples Quick Check
.
O
has radius 5. Point
P
and point
A
is on
O
such that
PA
is outside
O
such that
PO
= 12, = 13. Is
PA
tangent to .
O
at
A
? Explain.
Draw the situation described in the problem.
For
PA
to be tangent to
O
at
A
,
A
must be a right triangle, and
PO
2 =
PA
must be a right angle, 2 +
OA
2 .
OAP PO
2
PA
2 +
OA
2 Is
OAP
a right triangle?
12 2 144 13 2 194 + 5 2 Substitute.
Simplify.
Because
PO
2
PA
2 +
OA
2 ,
PA
is not tangent to
O
at
A
.
HELP GEOMETRY
Tangent Lines LESSON 12-1 Additional Examples
QS
and
QT
are tangent to
O
at points
S
and
T
, respectively. Give a convincing argument why the diagonals of quadrilateral
QSOT
are perpendicular.
Theorem 12-3 states
that two segments tangent to a circle from a point outside the circle are congruent.
Because
QS
and
QT
are tangent to
O
,
QS QT
, so
QS
=
QT
.
OS
=
OT
because all radii of a circle are congruent. Two pairs of adjacent sides are congruent. Quadrilateral
QSOT
is a kite if no opposite sides are congruent or a rhombus if all sides are congruent.
By theorems in Lessons 6-4 and 6-5, both the diagonals of a rhombus and the diagonals of a kite are perpendicular.
Quick Check HELP GEOMETRY
Tangent Lines LESSON 12-1 Additional Examples
XYZW
.
.
C
is inscribed in quadrilateral
XYZW
. Find the perimeter of
XU YS
=
XR
=
YR
= 11 ft = 8 ft
ZS WU
=
ZT
= 6 ft =
WT
= 7 ft By Theorem 12-3,
two segments tangent to a circle from a point outside the circle are congruent.
p
=
XY
+
YZ
+
ZW
+
WX
=
XR
+
RY
+
YS
+
SZ
+
ZT
+
TW
+
WU
+
UX
= 11 + 8 + 8 + 6 + 6 + 7 + 7 + 11 = 64 The perimeter is 64 ft.
Definition of perimeter
p
Segment Addition Postulate Substitute.
Simplify.
Quick Check HELP GEOMETRY