Download Unit 11 Section 3 Notes and Practice

Inscribed Angles and Tangents Notes and Practice
Inscribed Angles
An inscribed angle is an angle whose vertex is ON the circle and whose sides contain chords of the circle.
C
CBA is an inscribed angle.
Minor arc CA is the intercepted arc of CBA
B
A
There is a theorem and 3 corollaries that give us more information about the relationship between an
inscribed angle and a circle.
Theorem: The measure of an inscribed angle is half the measure of the intercepted arc.
C
mCBA 
B
1
m(arc CA)
2
A
For example: If the measure of arc CA is 110º, then mCBA  55 º.
Corollary 1: Two inscribed angles that intercept the same arc are congruent.
A
ADB intercepts arc AB; ACB intercepts arc AB
D
Therefore, ADB  ACB
B
C
Corollary 2: An angle inscribed in a semicircle is a right angle.
A
If AC is a diameter of Circle O, then
ABC is a right angle.
O
C
B
Corollary 3: The opposite angles of a quadrilateral inscribed in a circle are supplementary. (Remember an
inscribed polygon has every vertex of the polygon touching the circle.)
Examples
1. If m<ABC = 80°, find measure of arc AC.
2. If the measure of arc AB = 100° and the measure of arc BC = 170°, find m<ABC.
3. Find m<1 and m<2.
4. Find m<A and m<B.
D
Now, “You Try” these:
1. Find m<D and m<C if m<A = 75 and m<B = 60
A
.
C
B
2. Refer to figure 1. Find each measure.
a. m<ABC
b. mCD
c. mAD
d. m<BAC
e. m<BCA
f. mAB
Figure 1
Tangents
You have already learned about central angles in the circle, arcs and arc lengths. You will continue to learn about other
relationships of angles, lines and line segments in this unit.
A tangent to a circle is a line in the same plane as the circle that
intersects the circle at exactly one point.
is tangent to circle P.
The point where the tangent and circle intersect is called the point
of tangency. A is the point of tangency in this example.
is a radius that intersects with
a tangent segment.
.
is a tangent ray and
is
There are a couple of key theorems that relate a radius and a tangent to a circle. Can you hypothesize what the relationship
exists between these two key parts of the circle?
Theorem: If the tangent intersects with a radius that contains the
point of tangency, then the radius and tangent form a right angle.
And its converse: If a line in the same plane as a circle is
perpendicular to a radius at its endpoint on the circle, then the line is
tangent to the circle.
Example 1:
and
the value of x?
are radii of circle O.
and
are tangent to circle O at points A and C, respectively. What is
Solution: The angles at A and C are right angles
since
and
are tangents. ABCO is a quadrilateral
with angle measures whose sum is 360o.
90 + 123 + 90 + x = 360
303 + x = 360
x = 57
Example 2: Find the value of y when
is tangent to circle G.
Solution:
y + 90 + 47 = 180
y + 137 = 180
y = 43
Example 3: What is the length of the tangent to circle P?
Solution: PC and PA = 5 because they are radii of circle P, so
PB = 13 in length.
is a right angle, so
is the
hypotenuse of
Using Pythagorean Theorem,
25 +
= 144
In circle H to the right,
and
are radii, and therefore
congruent.
is the common side to
and
Both
triangles are right triangles.
and
are congruent by
Hypotenuse-Leg Theorem. Then we use CPCTC to identify that
.
Fact: Two segments from the same point tangent to the same circle
must be congruent.
We can use this theorem to find the perimeter of any polygon circumscribed around a circle with tangent segments.
Defnition: A polygon is circumscribed about a circle if the sides are tangent to the circle.
Example 4:
is circumscribed around circle P.
Solution: Perimeter of
,
, and
= MJ + MI + IL + LK + KN + NJ
= 6+6+8+8+9+9
= 46 units
Example 5: Find the perimeter of ABCD.
Solution: The two tangents that intersect
at C are each 4 units long (CW & CX).
Then XD is the remaining 3 units of DC.
XD = DY = 3 which leaves 5 units for YA.
YA = AZ = 5 and ZB = BW = 5. Therefore,
AB = 10 and BC = 9 and the perimeter =
AB + BC + CD + DA = 10 + 9 + 7 + 8 = 34.
Now, “You Try” the following practice problems. Answers to “You Try” problems can be found at the end of these notes.
1.
2.
is tangent to circle P.
Find the m<B.
and
are tangent to circle P.
What is the measure of the central angle P?
3. The distance between circles O and P is 8 units. How long is FG to the nearest tenth?
4. Find the perimeter of the pentagon circumscribed about circle P.
5. Find the radius of circle P.
6. Solve for x.
Answers to “You Try” problems.
1. 12o
2. 152o
3. 17.9
4. 36 units
5. CP = 7
6. x = 9