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Math 2 Module 7: Circles, A Geometric Perspective

7.1 Ready, Set, Go!

Questions 1 - 4

This video will show you how to find the scale factor of a scaling transformation: https://youtu.be/b4ORXN7a0-w

Questions 5 - 7

To solve this problem, we start drawing lines from similar points of both figures that intersect at a point called the center of dilation. This shows you how to find the center of dilation: https://www.youtube.com/watch?v=imPFvLS-gXY

Questions 8 - 10

A "center of rotation" is any point that's the same distance from both of the points that the two points could be rotated around and pass through each other's position. 

Questions 11 - 13

Find the perpendicular bisector of two pairs of corresponding vertices of an object and its image. Where the perpendicular bisectors intersect/converge is the center of rotation: https://www.youtube.com/watch?v=FrpkFL3yc3Y

Questions 14 - 16
The video tells about calculating area and circumference of circle: https://www.youtube.com/watch?v=O-cawByg2aA

7.2 Ready, Set, Go!

Question 1

The angles in a triangle always add up to 180 degrees. This video shows you how we use this angle rule to solve problems: https://www.youtube.com/watch?v=YnDeYhBoCzs

Question 2

We start with two math rules: the angles inside a triangle must add to 180 degrees and any line has an angle of 180 degrees. This video shows you how we use these two angle rules to solve problems: https://www.youtube.com/watch?v=EZ6dOlRQDBo

Question 3

The angles in a triangle always add up to 180 degrees. This video shows you how we use this angle rule to solve problems: https://www.youtube.com/watch?v=YnDeYhBoCzs

Question 4

This video introduces the Triangle Angle Bisector Theorem: https://www.youtube.com/watch?v=6WZRypy7ry0

Questions 5 - 6

To find the smallest rotation symmetry in a regular polygon, divide 360 by the number of sides: https://www.youtube.com/watch?v=vIvU5j-Crx8

Questions 7 - 9

Two figures are said to be similar if the corresponding angles are congruent (equal). Note that similarity does not imply that the length of the sides are equal but does imply they are proportional: https://www.youtube.com/watch?v=LsIDYhcUJhU

Questions 10 - 11

The line of reflection is found out by matching coordinates of both figures. The perpendicular bisector of these lines is the line of reflection: https://www.youtube.com/watch?v=435Ipnmp7bA

Questions 12 - 13

Think about how you can use the chords to find the center of a circle. Think about the perpendicular bisectors of each chord. Draw them on the diagram: https://www.youtube.com/watch?v=kGz_7OcU4EM

7.3 Ready, Set, Go!

Questions 1 - 6

The angle of rotational symmetry can be found out from the number of sides of the polygon. This video will help you find the angle of rotational symmetry: https://youtu.be/vIvU5j-Crx8

Think about the different ways you could fold each shape in half and end up with two figures that completely overlap. This video will help you find the lines of reflection: https://youtu.be/vx-9vd5BBxI

Questions 7 - 10

Find the remaining sides and angles with the help of trigonometry functions. This video will walk you through similar examples: https://www.youtube.com/watch?v=2b-qB_LLofY

Questions 11 - 12

Find inscribed arc and inscribed angle with the help of theorem of inscribed angle: https://www.youtube.com/watch?v=sO1JgLZgP6Q

Questions 13 - 14

Given arc and angle find other angles: https://www.youtube.com/watch?v=_5IJDsX8d9Y

Question 15

Proving that an inscribed angle is half of a central angle that subtends the same arc. This video will give you some intuition about the proof: https://www.youtube.com/watch?v=MyzGVbCHh5M

 

Questions 16 - 17

This video will teach you how to find the length of an arc and the area of a sector: https://www.youtube.com/watch?v=D9xX8ip7kRc

7.4 Ready, Set, Go!

Questions 1 - 6

Radius is given, from that find area and circumference according to the the formula: https://www.youtube.com/watch?v=zHkAEbkuqLg

Questions 7 - 12

This video will teach you how to calculate the perimeter of regular polygons: https://youtu.be/huUr6ietSG4 

 

This video will help you find the measure of interior angles of regular polygons: https://youtu.be/Cjhpn4VrfMU   

This video will help you find the area of a regular polygon: https://youtu.be/z50C1b2sO5M

Questions 13 - 16

Given angle and radius, using formula of sector find the area: https://youtu.be/7f9_U85_YX4

7.5 Ready, Set, Go!

Questions 1 - 6

This video will show you how to do conversions using inches, feet, yards, and miles: https://youtu.be/bcM5J-wYHjE

Questions 7 - 12

Use the equation C = (5/9)(F - 32) with F = 10 to convert to Celsius. The video will provide a few examples to show you how to do this: https://youtu.be/NvjfJfGhCxs

Questions 13 - 16

For this problem, you need to use circle geometry rules to find unknown values (and figure out which ones you can't find with the information given). This video reviews the circle theorems: https://www.youtube.com/watch?v=XUus6-9E9sQ

Questions 17 - 18

What individual shapes do you see? How can you calculate the perimeter of those shapes? This video will walk you through multiple examples on how to solve similar problems: https://youtu.be/_e7j6rE7_Pg

7.6 Ready, Set, Go!

Questions 1 - 6

This video shows you how to find the missing terms in an arithmetic sequence: https://www.youtube.com/watch?v=g7Tl3clSDYw

Questions 7 - 9

I found a video that shows you how to find the recursive formula of an arithmetic sequence given two terms. The example uses different notation but don't let that scare you! His a₃ is the same as f(3)). https://www.youtube.com/watch?v=VguyPrGaXLk

Questions 10 - 12

To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

 

When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

Questions 13 - 16

This question asks you to evaluate the exponential function they've given you. This video shows you how to evaluate exponential functions: https://www.youtube.com/watch?v=GQIcZeKR9RI

Question 17

Look at the parentheses in question 15, how do they change the effect of the exponent? https://www.youtube.com/watch?v=pJjcqlGUPg4

Questions 18 - 19

These questions asks you to evaluate the linear function they've given you. This video shows you how to evaluate linear functions: https://www.youtube.com/watch?v=GQIcZeKR9RI

7.7 Ready, Set, Go!

Questions 1 - 6

This video will show you how to write ratios: https://www.youtube.com/watch?v=puku5vUCOcE

Questions 7 - 8

This video will show you how to find the circumference and area of a circle: https://www.youtube.com/watch?v=zHkAEbkuqLg

Question 9

Set up the equation for the area of a sector, plug in the known values, and then solve for the missing variable. This video will show you how to do this: https://youtu.be/3HvtFRHZP20

Questions 10  - 11

Set up the equation for the arc length, plug in the known values, and then solve for the missing variable. This video will show you how to do this: https://youtu.be/QwuwTIHRx7E

Question 12

Set up the equation for the area of a sector, plug in the known values, and then solve for the missing variable. This video will show you how to do this: https://youtu.be/3HvtFRHZP20

Questions 13  - 18

Find the area of both figures. The area of the darker region can be found by subtracting the area of the polygon from the area of the circle. This video will walk you through a similar problem: https://youtu.be/Sbg5qht02ko

The point is, if you know the arc length and want to find the area, you should first figure out the radius, and from there you can calculate the area.

7.8 Ready, Set, Go!

Questions 1 - 4

This video shows you how to find the common ratio for a geometric sequence: https://youtu.be/-rofZ1OtTik

Questions 5 - 8

Start by looking at the pattern and decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

 

This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

 

When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

Questions 9 - 12

You need to find the equation of the line on the graph. This video has an example of a line with a positive slope: https://www.youtube.com/watch?v=mmWf_oLTNSQ

 

And this video is an example of a line with a negative slope: https://www.youtube.com/watch?v=dBFbFMvEwQg

7.9 Ready, Set, Go!

Questions 1 - 4

This video will show you how to find the scale factor of a scaling transformation: https://youtu.be/b4ORXN7a0-w

Questions 5 - 7

To solve this problem, we start drawing lines from similar points of both figures that intersect at a point called the center of dilation. This shows you how to find the center of dilation: https://www.youtube.com/watch?v=imPFvLS-gXY

Questions 8 - 10

A "center of rotation" is any point that's the same distance from both of the points that the two points could be rotated around and pass through each other's position. 

 

This video shows you how to find the center of rotation for pairs of shapes on a graph: https://www.youtube.com/watch?v=FrpkFL3yc3Y

Questions 11 - 13

Find the perpendicular bisector of two pairs of corresponding vertices of an object and its image. Where the perpendicular bisectors intersect/converge is the center of rotation: https://www.youtube.com/watch?v=FrpkFL3yc3Y

Questions 14 - 16
The video tells about calculating area and circumference of circle: https://www.youtube.com/watch?v=O-cawByg2aA

7.10 Ready, Set, Go!

Questions 1 - 3

This video will show you how to find the center of a circle with chords: https://www.youtube.com/watch?v=kGz_7OcU4EM

Questions 4 - 5

This video will explain to you the relationship between a central and an inscribed angle: https://youtu.be/MyzGVbCHh5M

Questions 6 - 13

This video shows you how to find the volume and surface area of a rectangular prism: https://www.youtube.com/watch?v=2B_N6k0o3-M

Remember to multiply each side of the prism by 4, draw the enlarged prism, and then find the surface area and volume of the new prism. This video shows you how to find the volume and surface area of a rectangular prism: https://www.youtube.com/watch?v=2B_N6k0o3-M

Questions 14 - 22

This video will show you how to convert degrees to radians and vice versa: https://youtu.be/O3jvUZ8wvZs

Questions 23 - 24

This video will show you how to find the area of a sector: https://youtu.be/7f9_U85_YX4

Questions 25 - 26

This video will show you how to find the length of an arc: https://youtu.be/v-ZV_jMumFE

7.11 Ready, Set, Go!

Questions 1 - 6

This video will show you how to find the distance between two points: https://www.youtube.com/watch?v=nyZuite17Pc

Questions 7 - 10

Remember to multiply the height and base dimensions by the scale factor, draw the new pyramid, and then find its surface area and volume.

 

This video shows you how to find the volume and surface area of a pyramid: https://www.youtube.com/watch?v=OY2RRlPPT7A

Questions 11 - 12

This video will show you how to find the volume and surface area of a cone: https://www.youtube.com/watch?v=JN3LhO0YXw8

Questions 13 - 16

This video will show you how to find the missing sides and angles in triangles using trigonometry: https://www.youtube.com/watch?v=Jsiy4TxgIME

7.12 Ready, Set, Go!

Questions 1 - 3

This video shows you how to multiply binomials: https://youtu.be/ZMLFfTX615w

Questions 4 - 6

This video will show you how to factor quadratic expressions into two binomials: https://www.youtube.com/watch?v=u1SAo2GiX8A

Question 7

This video shows you how to find the area of a parallelogram: https://www.youtube.com/watch?v=s8OPoot0IUg

Question 8

We know AD and HD are parallel lines, but we don't have enough information to be sure of anything else.

Question 9

Part a) to find the volume, first find the length of each side in terms of a and b. For example, the width is b+a. To find the volume of a rectangular prism, check out this video: https://www.youtube.com/watch?v=u1nWI2b0fT4 This is also relevant to part d)

Part b) if you pull the block apart along the cut lines, how many pieces will you have in total?

Questions 10 - 12

What is the difference between similarity and congruence? This video will teach you the difference between similarity and congruence: https://www.youtube.com/watch?v=I8IOMUGRmDg

This video will show you how to compare shapes to see if they are similar or congruent. Remember that if they are congruent, the scale factor will be 1: https://youtu.be/03-G60uEeQs

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