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Ch 12: Circles
12‐1 Tangent Lines
12‐2 Chords and Arcs
12‐3 Inscribed Angles
12‐4 Angle Measures and Segment Lengths
12‐5 Circles in the coordinate plane
12‐1 Tangent Lines
Focused Learning Target: I will be able to
Use the relationship between a radius and a tangent Use the relationship between two tangents from one point
CA Standard(s): Geo 7.0 Geo 21.0
Vocabulary:
Tangent to a circle
Point of tangency
Inscribed in Circumscribed about
Example 1: Finding angle measures: I’ll do one: We’ll do one together: You try one:
ML and MN are tangent to circle O. Find the value of x
2
Example 2: Real‐World Connection I’ll do one: We’ll do one together: You try one:
Example 3: Finding a Tangent I’ll do one: We’ll do one together: You try one:
Determine whether a tangent line is shown. Explain.
3
Example 4: Using Theorem 12‐3 I’ll do one: We’ll do one together: You try one:
Example 5: Circles Inscribed in Polygon I’ll do one: We’ll do one together: You try one:
Circle O is inscribed in triangle PQR. Triangle PQR has a perimeter of 88 cm. Find QY
Circle O is inscribed in triangle ABC. Find the perimeter of triangle ABC.
4
12‐2 Chords and Arcs
Focused Learning Target: I will be able to
Use the relationship between a radius and a tangent
Use the relationship between two tangents from one point
CA Standard(s): Geo 2.0 Geo 7.0 Geo 21.0
Vocabulary:
Chord:
Example 1: Using Theorem 12‐4 I’ll do one: We’ll do one together: You try one:
List what you can conclude from the diagram.
Example 2: Using Theorem 12‐5 I’ll do one: We’ll do one together: You try one:
What is the value of a in the circle?
Find the value of x in the circle
5
Example 3: Using Diameters and Chords I’ll do one: We’ll do one together: You try one:
Find the missing lengths to the nearest tenth
____________________________
Use the circle below:
a. Find the length of the chord. b. Find the distance from the midpoint of the chord to the midpoint of its minor arc.
Find the value of x to the nearest tenth.
6
12‐3 Inscribed Angles
Focused Learning Target: I will be able to
Find the measure of an inscribed angle
Find the measure of an angle formed by a tangent and a chord
CA Standard(s): Geo 2.0 Geo 7.0 Geo 21.0
Vocabulary:
Inscribed angle Intercepted arc
In the circle at right, the vertex of C is on circle O and the sides of C are chords of the circle. C is an inscribed angle. Arc AB is the intercepted arc of C .
I’ll do one:
Find the values of the variables.
We’ll do one:
Find the values of the variables.
You try:
Find the values of the variables.
7
I’ll do one: We’ll do one: You try:
Find the measure of the numbered angle.
Find the measure of the numbered angle.
Find the measure of the numbered angle.
I’ll do one: We’ll do one: You try:
Find the values of x and y.
Find the values of each of the variables.
Find the values of x and y.
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12‐4 Angle measures and Segment Lengths Focused Learning Target: I will be able to
Find the measures of angles formed by chords, secants, and tangents
Find the lengths of segments associated with circles
CA Standard(s): Geo 2.0 Geo 7.0 Geo 21.0
Vocabulary: Secant: a line that intersects a circle at two points.
AB
is a secant ray, and AB is a secant segment.
Finding Angle Measures: I’ll do one:
Find the value of x:
Find the value of y:
Find the value of y:
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Finding Segment Lengths:
It does not matter which point you choose, or which lines you use, the product (PA)(PB) remains constant.
I’ll do one:
Find the value of the variable. If the answer is not a whole number, then round to the nearest tenth.
We’ll do one together:
Find the value of the variable. If the answer is not a whole number, then round to the nearest tenth.
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We’ll do another one:
Find the value of the variable. If the answer is not a whole number, then round to the nearest tenth.
You do one:
Find the value of the variable. If the answer is not a whole number, then round to the nearest tenth.
12‐5 Circles in the Coordinate Plane
Focused Learning Target:
Write an equation of a circle
Find the center and radius of a circle
CA Standard(s): Geo 7.0 Geo 17.0
Vocabulary:
Standard form of an equation of a circle
The way the equation of a circle is currently written is in standard form.
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Example 1: Writing the equation of a circle. I’ll do one: We’ll do one together: You try one:
Write the standard equation of the circle with center (5, ‐2) & radius 7.
Write the standard equation of the circle with center (‐2, ‐1) & radius
2 .
Write the standard equation of the circle with center (3, 5) & radius 6.
Example 2: Writing the equation of a circle given 2 points I’ll do one: We’ll do one together: You try one:
Write the standard equation of the circle with center (1, ‐3) and passes through the point (2, 2).
Write the standard equation of the circle with center (2, 3) and passes through the point (‐1, 1).
Write the standard equation of the circle with center (7, ‐2) and passes through the point (1, ‐6).
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Example 3: Writing the equation of a circle given the graph I’ll do one: We’ll do one together: You try one:
Write the standard equation of the circle.
Write the standard equation of the circle.
Write the standard equation of the circle.
Example 4: Finding the center and radius of a circle
I’ll do one: Find the center and radius of the circle with equation
64)2()4( 22 yx . Then graph the circle.
x
y
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We’ll do one together: Find the center and radius of the circle with equation
100)3()2( 22 yx . Then graph the circle.
x
y
You try one: Find the center and radius of the circle with equation
25)1()4( 22 yx . Then graph the circle.
x
y