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Geometry Chapter 12 Notes
Warm Up #23: Review of Circles
/ 1.) A central angle of a circle is an angle with its vertex at the ______of the circle. Example: ______2.) An arc is a section of a circle. Examples: ______, ______
3.) We measure arcs in ______Example:
4.) A minor arc is a section of a circle that is less than ______.
Example of a minor arc: ______
5.) A major arc is a section of a circle that is greater than _____.
Example of a major arc: ______
6.) The entire arc of a circle measures ______
7.) A semicircle is ______of a circle (formed by a ______) and measures ______.
8.) Examples of a semicircle: ______, ______/
The measure of an arc is equal to the measure of its ______angle.
/ Arc Addition Postulate
The sum of the measures of two adjacent arcs is equal to the measurement of the larger arc that they form together.
Find m<1:
9.)
/ 10.)
/ 11.)
Notes #23: Tangents (Sections 12.1)
Tangents/ 1.) A ______is a segment, ray, or line that touches a circle only once. Name four tangents: ______, ______, ______
2.) The point where a tangent touches a circle is called the ______.
3.) Name the point of tangency: ____
Theorem 12.1
A tangent is ______to the ______drawn to the point of tangency.
Draw this relationship to the left.
Use this theorem to complete:
For #4-5, assume that the lines that appear to be tangent are tangent.
4.) If x = 100°, find
5.) If find x /
For #6-8, is tangent to the circle at T
6.) If OT = 5, JT = 10, JO = ______
7.) If and JO = 12, JT =_____
8.) If JK = 18 and KO = 7, then JT = ____
Theorem 12-3
Tangents to a circle from a common point are ______.
Use this corollary to complete:
9.) AX = 5, DX = ______
/ 10.) Find the perimeter of
11.) Find the perimeter of the quadrilateral:
12.) We say that a polygon is ______a circle when all vertices (corners) are on the circle. In this case, we can also say that the circle is ______about the polygon.
13.) Describe this figure in two ways:
/ 14.) Describe this figure in two ways:
15.) Draw a circle inscribed in a square. / 16.) Draw a circle circumscribed about a pentagon.
Compass Practice: (When using your compass, you must keep the point(called the center) still.)
17.) Use your compass to make small 18.) Use your compass to make a large circle.
circles. Label their centers.Label the center.
19.) Use your compass to draw arcs of a circle.
Construction #1: Given a segment, construct a segment congruent to the given segment (pg. 44)
Steps:1.) Using your straightedge, draw a segment longer than . Pick a point on this segment and label it X. [Our goal is to “measure” and mark (or cut and paste)onto our new segment.]
2.) Set the width (or radius) of your compass to the length of . This means place the point of your compass on A, and then stretch your compass so that the pencil is on point B. [You have now “stored” this length into your compass.]
3.) Without changing the opening of your compass, place its point on X. Use the compass to mark your segment with an arc. Label this intersection as Y.
4.) Check with a ruler that /
Construct a segment congruent to each given segment:
20.) / 21.)Warm Up #24:
1.) Find the perimeter of the quadrilateral:/ 2.) is tangent to circle A at B. If BC = 12 and , find the length of the radius and the diameter of the circle.
3.) Find the area of a regular hexagon with perimeter 96m. / 4.) Find the volume of a cylinder with diameter 8in and height 10in.
Notes #24: Section 12.2 and Constructing Congruent Angles
Chords (Section 12.2)
1.) are ______. These segments connect any ____ points on a circle.2.) What is a name for the longest chord in a circle? ______
3.) are ______. These are lines that contain a ______. /
/ Theorem 12.4
(1) Congruent arcs have congruent ______.
(2) Congruent chords have congruent ______.
4.)
/ 5.)
Theorem 12.6
A diameter or radius that is ______to a chord bisects the ______and its ______.
Use this theorem to complete: (look for right triangles!!) /
6.) XY = ____, Diameter = ____
/ 7.) PQ = 16, OM = ______, Diameter = ______
/ Theorem 12.5
(1) Chords that are ______from the center are ______
(2) ______chords are ______from the center
**remember that distance is measured with a perpendicular segment, so look for right angles and ways to make righttriangles**
Use this theorem to complete:
8.) OM = ON = 6, CM = 8, EF = ___, radius = ___, diameter = ___
Construction #2: Construct an angle congruent to the given angle. (pg. 45)
Steps:1.) Using your straightedge, draw a segment. (This will be the “base edge” of your angle.) Label the left endpoint of the segment E. (This will be the vertex of your new angle.)
2.) Set the width (or radius) of your compass to a length shorter than .
3.) Place the center (or point) of your compass at point B and draw an arc that passes through both sides of the original angle. Mark these two intersection points as X and Y.
4.) Without changing the width of your compass, draw this same arc with the compass point at E. Mark the intersection point on the base as M.
5.) Re-set your compass to the distance between intersection points X and Y. (This measures the width of the original angle.)
6.) Without changing the width of your compass, put the point of your compass on M and draw an arc. Mark this intersection as D.
7.) Connect points D and E. Check with your protractor that /
Construct an angle congruent to each given angle:
9.)/ 10.)
Warm Up #25:
1.) OT = 6, RS = 8, OR = ______,radius = _____, diameter = _____
/ 2.) CD=12, OM=8, ON=4, EF = ___,
radius = ___, diameter = ___
3.) Solve for x:
Notes #25: Spheres, Inscribed Angles, Constructing Perpendiculars (Section 12.3)
A plane passes h cm from the center of a sphere with radius R, forming a circle with radius r. Find the indicated values:
1.) R = 5, h = 4, r = ?2.) r = 9, h = 12, R = ?
Special Cases: / An inscribed angle is an angle whose vertex is _____ the circle.
Example: ______
The measure of an inscribed angle equals ______the measure of the intercepted arc.
Example: = ______, = ______
An inscribed angle that intercepts a
semicircle is ____ Example: ______
/ The measure of an angle formed by a tangentand a chord equals ______the measure of its intercepted arc. Example: ______
If two inscribed angles intercept the same arc, then the angles are ______.
Example: ______/
/ If a quadrilateral is inscribed in a circle, then its opposite angles are ______
Example: ______
Solve for the indicated variables:
3.) x = ____, y = _____
/ 4.) x = _____, y = _____
5.) x = ____, y = _____
/ 6.) x = ____, y = _____
7.) x = ____, y = ____, z = _____
/ 8.) x = ____, y = ____,z = _____
Constructing Perpendicular Lines
Construction #3: Given a point on a line, construct the perpendicular to the line at the given point (pg. 182)
1.) Given point C on line k2.) Using C as your center, draw two arcs on line k, one arc to the left of C and one to the right of C.
3.) Label these new points as X and Y.
4.) Place the center of your compass on X and using a radius larger than CX, draw one arc above line k and one arc below line k.
5.) Do the same from point Y. Both pairs of arcs must intersect above and below line k. Label these points M and N.
5.) Connect M and N. This segment should intersect point C and be perpendicular to line k. /
Construct a perpendicular through the indicated point:
9.) 10.)
Construction #4: Given a point outside a line, construct the perpendicular to the line from the given point (pg. 183)
Steps:1.) Place the center of your compass on point P.
2.) Using a consistent width, draw two arcs that intersect line k. Label these points X and Y.
3.) From X, draw an arc below line k
4.) From Y, draw an arc below line k
5.) These two arcs must intersect. Label this intersection Z.
6.) Connect P and Z; this segment is perpendicular to line k. /
Construct the perpendicular segment from P through line k.
11.)12.)
Warm Up #26: Quiz Review
1.) Find the perimeter of the quadrilateral/ 2.) is tangent to circle A at B. If BC = 18 and , find the length of the radius and the diameter of the circle.
3.) Name one of each:
a) central angle
b) chord
c) secant
d) inscribed angle / / 4.) OT = 12, ST = 16, OR = ______,
radius = _____, diameter = _____
5.)
/ w = ____
x = _____
y = _____
z = _____
Warm Up #27: Solve for the variables
1.)/ 2.)
3.)
/ 4.) CD=12, OM=3, ON=, EF = ___,
radius = ___, diameter = ___
Notes #27: Angle Measures and Segment Lengths (Section 12.4), Review of Constructions
Interior and Exterior Angles
Solve for x: / Theorem 12.11 (Part I)
An angle formed by two chords is equal to ______the ______of its two intercepted arcs.
1.)
/ 2.)
/ 3.)
Theorem 12.11 (Part II)
An exterior angle formed by (a) two secants, (b) two tangents, or (c) one tangent and one secant is equal to ______the ______of its two intercepted arcs.
(a)
/ (b)
/ (c)
Solve for x and y:
4.)
/ 5.)
6.)
/ 7.)
Complete the constructions:
8.) Construct a segment congruent to the given segment:/ 9.) Construct an angle congruent to the given angle:
10.) Construct a line perpendicular to the given line through the given point:
/ 11.) Construct a line perpendicular to the given line through the given point:
12.) Construct a line perpendicular to the given line through the given point:
/ 13.) Construct a line perpendicular to the given line through the given point:
14.)Construct an angle congruent to the given angle:
Warm Up #28:
Circle Review
1.) Find OM and MQ/ 2.) Find OM and MQ
3.) Find x and y
/ 4.) Find x and y
Notes #28: Segment Lengths, Constructing Parallel Linesand Perpendicular Bisectors (Section 12.4)
Segment Lengths in Circles
Solve for x: / Theorem 12.12 (Part I)
When two chords intersect in a circle, the product of the ______of one chord equals the product of the ______of the other chord
(piece)(piece) = (piece)(piece)
1.)
/ 2.)
Theorem 12.12 (Part II)
When (a) two secants or (b) one tangent and one secant
are drawn to a circle, then:
(exterior segment)(whole length) = (exterior segment)(whole length)
(a)
/ (b)
Solve for the variables:
3.)
/ 4.)
5.)
/ 6.)
7.)
/ 8.)
9.)
Construction #5: Given a point outside a line, construct the parallel to the given line through the given point (pg. 181)
Steps:1.) Draw points A and B, in that order and not too far apart, on line k.
2.) Draw and extend this ray a considerable distance. (At P, we will construct an angle congruent and corresponding to .)
3.) Set your compass to the width of AB. With the center of your compass at A, draw an arc that passes through line k and . Mark the new point of intersection as X.
4.) Without changing the width of your compass, put its center at P and draw a similar arc above and to the right of point P. Mark the point where this arc and intersect as M.
5.) Change the width of your compass to be the length XB.
6.) Keeping this width, put the center of your compass on M and draw an arc that intersects the previous arc. Mark this point of intersection as N.
7.) Draw . This line is parallel to line k. /
Construct the parallel linetok through P.
10.) 11.)
Construction #6: Construct a perpendicular bisector to a given segment (pg. 46)
Steps:1.) Set the width (or radius) of your compass to a length longer than half the given segment.
2.) With the center of your compass on point A, draw one arc above and one arc below the segment.
3.) With the center of your compass on point B, draw one arc above and one arc below the segment. 4.) Make sure that you find where these pairs of arcs intersect. Mark these points as X and Y.
5.) Connect X and Y; this is your perpendicular bisector. /
12.) 13.)
/ ______Warm Up #29: Circle Review
Solve for the variables:
1.)/ 2.)
3.)
/ 4.)
Notes #29: Circles in the Coordinate Plane, Constructing Angle Bisectors
(Section 12.5)
CirclesEquations of circles are written in this form:
Where (h, k) is the ______of the circle and r is the ______of the circle
Graph and find the equation of each circle:
- Graph the center point
- From this point, go the distance of the radius up, down, left, and right
- Connect these 4 points as a circle
1.) C (-2, 4) r = 4
2.) C(5, 0) , r = 3 /
Name the center and radius of each circle, then graph.
3.)4.) /
Write the standard equation of each circle:
- Find the center point
- From this point, count the distance to the highest point on the circle. This is the radius.
5.)
/ 6.)
Write the standard equation of the circle with the given center that passes through the given point:
- Plug the center point into the standard equation as (h, k)
- Plug the second point into that equation as (x, y)
- Solve for r. Plug into equation from first step.
7.) A circle has a center (1, 3) and passes through the point (4, -1). Find the equation of this circle and graph it. / 8.) Find the equation of the circle whose center is (-3, -2) and goes through the point (-1, 2).
Construction #7: Construct the bisector of a given angle (pg. 47)
Steps:1.) Set the width (or radius) of your compass to a length shorter than .
2.) Place the center (or point) of your compass at point B and draw an arc that passes through both sides of the original angle. Mark these two intersection points as X and Y.
3.) Keeping the same width, place the center of your compass at point X and draw an arc in the interior of the angle. Do the exact same process with your compass at point Y.
4.) Label the intersections of these two interior arcs Z
5.) Using your straightedge, draw
6.) bisects /
Using only a compass and a straightedge, bisect the given angles:
9.)10.)
HW #30 Geometry Chapter 12 Study Guide Name : ______
Necessary Skills: Pythagorean Theorem and Special Right Triangles. Solve for x and y
1.)/ 2.)
/ 3.)
/ 4.)
Key Vocabulary: Name one example of each /
5.)
a. radius
c. chord
e. tangent
g. central angle
i. minor arc / b. diameter
d. secant
f. point of tangency
h. inscribed angle
j. major arc
6.) Draw each:
a. A rectangle inscribed in a circle
Tangents: / b. A circle circumscribed about an obtuse triangle.
7.) Complete: JT is tangent to the circle at T
a.) If OT = 4, JO = 12, then JT = ______
c.) If JK = 4 and KO = 6, then JT = ______/ b.) If and JT= 12,
JO = ____, OT = ____
Central and Inscribed Angles:Solve for x and y
8.)/ 9.)
/ 10.)
11.)
/ 12.)
/ 13.)
14.)
/ 15.)
/ 16.)
Chords and Arcs:
17.)
/ 18.)OT = 12, RS = 4,
OR = ___, radius = ___, diameter=___
/ 19.) CD = 6, OM = 4, ON = 3, EF = ___,
radius = ___, diam. = ___
/ 20.)EF = 8, ON = 3, OM = 2, CD = _____
Interior and Exterior Angles:Solve for x
21.)/ 22.)
/ 23.)
24.)
/ 25.)
/ 26.)
Segments and Lengths:Solve for x
27.)/ 28.)
/ 29.)
30.)
/ 31.)
/ 32.)
Find the perimeter of each polygon:
33.)/ 34.)
Complete the problems about circles:
35.) Find the equation of the circle with center at (2, -7) and radius 5. / 36.) Find the equation of the circle whose center is (-3, -1) and goes through the point (1, 2)37.) Find the center and the radius of the circle described by: (x - 5)2 + (y +3)2 = 9. Graph.
/ 38.) Find the equation of the circle shown here:
39.) A chord is 4m from the center of a circle. If the radius of the circle is 10m, how long is the chord? / 40.) A sphere is sliced by a plane 2ft from the center of the sphere. If the radius of the circular plane made by the plane’s slice is 8ft, what is the radius of the sphere?
**Don’t forget worksheet problems 40-48**