Middle School Geometry Review

Unit 1:
Polynomial Expressions

Coefficient, Variable, Term

A Coefficient is a number used to multiply a variable: EX 5x², 5 is the coefficient.

A Variable is a letter or symbol used to stand in for a number we don't know. EX 4a, a is the variable

A Term is a number, a variable or the product of numbers and variables. EX These are all terms: 7, x, ac³, 4pqr

Expression, Standard Form, Leading Coefficient, Constant Term

An Expression is a collection of terms that are added or subtracted. EX These are expressions: 5x, 3a + 4, 7x²-2x+9

An Expression is in Standard Form if the term with the highest degree (exponent) is first and the other terms are in decreasing order by degree.

The Leading Coefficient is the coefficient of the first term when written in Standard Form.

A Constant Term is a number without a variable.

​Box Method

Multiple Choice

(2x + 5y - z) + (-6x - 4y + 7z)

• -4x +6y + 6z

• 4x + y + 6z

• -4x - y + 6z

• -4x + y + 6z

Multiple Choice

Simplify
(4x³ -3x² + 6x - 4) - (-2x³ + x² -20)

• 4x² - 6x³ - 2x + 6
• 3x + 5x² + 1
• 6X³ -4X² + 6X -2
• 3x + 2

Multiple Choice

• 2x² +19x +45

• 2x² + x + 45

• 2x² + x - 45

• 6x - 8

Multiple Choice

What is the perimeter of the triangle shown?

• 9r²+r-5

• 9r⁴+r²-4

• 9r²-r+4

• 9r²+r-4

Unit 2:
Geometric foundations, constructions & proofs

Multiple Choice

Which of the following constructions is illustrated?

• An angle is congruent to a given angle
• The bisector of a given angle
• The bisector of a given segment
• The perpendicular bisector of a given segment.

Multiple Choice

Name the construction.

• Angle bisector
• Perpendicular line to a point not on a line
• Perpendicular line to a point on the line
• Perpendicular bisector of a line segment

Multiple Choice

Name the construction.

• Angle bisector
• Perpendicular line to a point not on a line
• Angle median
• Perpendicular bisector of a line segment

Interior v. Exterior

Interior means BETWEEN the parallel lines

Exterior means OUTSIDE the paralell lines

Consecutive v. Alternate

Consecutive means on the SAME SIDE of the TRANSVERSAL

Alternate means on DIFFERENT SIDES of the TRANSVERSAL

​If 2 parallel Lines are cut by a transversal, then...

• ​alternate interior angles are congruent

• ​alternate exterior angles are congruent

• ​corresponding angles are congruent

• ​same-side (consecutive) interior angles are supplementary

Multiple Choice

Angle C and P are...

• Alternate Interior Angles
• Consecutive Interior Angles
• Corresponding Angles
• Alternate Exterior Angles

Multiple Select

Select all the vertical angles

• $$\angle J\ \&\ \angle K$$

• $$\angle G\ \&\ \angle I$$

• $$\angle K\ \&\ \angle E$$

• $$\angle E\ \&\ \angle H$$

Multiple Choice

In the diagram, which of the following statements are false?

• ∠1 and ∠4 are vertical angles
• ∠4 and ∠5 are alternate interior angles
• ∠1 and ∠8 are alternate interior angles
• ∠2 and ∠6 are corresponding angles

To prove CPCTC:

First, we need to prove that the two triangles are congruent with the help of any one of the triangle congruence criteria.

In the figure, determine how you could prove the triangles congruent.

SSS, SAS, AAS, ASA, HL

​

Multiple Choice

What reason could prove the triangles congruent?

• SSS

• SAS

• AAS

• ASA

• HL

Multiple Choice

What congruency postulate proves the triangles congruency?

• SAS

• ASA

• SSS

• HL

Multiple Choice

Are the triangles congruent, if yes, why?

• SSS

• ASA

• HL

• Not Congruent

Match

Match the triangles to the congruence theorem.

SSS

AAS

SAS

ASA

• SSS

• AAS

• SAS

• ASA

Multiple Choice

What is the "reason" for step 5 of the proof?

• Angle Bisector Theorem
• Reflexive property
• CPCTC Theorem
• Proof

Multiple Choice

What is the "statement" for step 2 of the proof?

• AD=AD
• AD=DA
• HD=DN
• HA = AN

Multiple Choice

What is the "reason" for step 3 of the proof?

• Vertical Angle Theorem
• given
• reflexive property
• proof

Multiple Choice

What is the length of

$$\overline{AD}$$  ?

• 3

• 5

• 6

• 11

Multiple Choice

If PONS is a rectangle, PN = 2x - 5 and OS = 17. Solve for the value of x.

• 6

• 11

• 12

• 22

Unit 3/4:
Transformations, congruency and similarity

Multiple Choice

Describe the translation algebraically

[right 3, down 2]

• (x+3, y-2)

• (x-3, y+2)

• (x+2, y-3)

• (x-2, y+3)

Multiple Choice

Describe the reflection algebraically. 

Hint: Look at the coordinates from ABC to A'B'C'

• (-x,y) 

• (x,-y) 

Multiple Choice

SQR rotated ______________ to get S'Q'R'

• 90^o clockwise

• 180^o clockwise

• 270^o clockwise

​ABC is mapped to A'B'C

Multiple Choice

What is the scale factor of this reduction?

Hint: Compare A'B' to AB

What do you multiply by to change the side length?

• 1/2

• 1/3

• 3

• 4

Multiple Choice

If A(2,1) is enlarged to A'(4,2),

how can this be described algebraically?

• (1/2x, 1/2y)

• (2x, 2y)

• (1/4x, 1/4y)

• (4x, 4y)

​Dilation

​(englargment/reduction)

​Rigid Transformations

​Translation, Rotiation, Refection

Multiple Choice

Which of the following describes the sequence of transformations shown?

• Reflect across the x-axis and then rotate -90 degrees around the origin
• Rotate -90 degrees around the origin and then translate down.
• Reflect across the x-axis and then reflect across the y-axis.
• Translate up and then rotate 90 degrees about the origin.

Multiple Choice

What does A(-3, 4) become after rotated 90 degrees counterclockwise?

• (-4, -3)
• (4, 3)
• (3, -4)
• (-4, 3)

Multiple Choice

Under what composition of transformations does triangle ABC map onto triangle A''B''C''

• Reflection over x-axis, then Rotation of 90 Degrees

• Rotation of -90 Degrees, then Reflection over y-axis

• Reflection over y-axis, then Rotation of -90 Degrees

• Rotation of 90 Degrees, then Reflection over x-axis

Multiple Choice

What is the sequence of transformations?

• Reflect over the y then reflect over the x
• Reflect over the x then reflect over the y
• Translate 4 units then rotate 90˚
• Rotate 90˚ then reflect over the x

Multiple Choice

Which triangle congruence theorem can be used to prove the triangles are congruent?

• SSS

• SAS

• ASA

• NONE

Similarity Conditions

​Angle Angle

​

Side-Angle-Side​

​

Side-Side-Side​

​

If you have 2 angles that are the same, triangles are similar.

​

If you have 2 sides that are proportional and the angle between them is congruent, triangles are similar.​

​

If you have 3 sides that are proportional, triangles are similar.​

​

Some text here about the topic of discussion

Multiple Choice

State if the triangles in each pair are similar. If so, state how you know they are similar.

• Yes, AA Similarity

• Yes, SSS Similarity

• Yes, SAS Similarity

• Not Similar

Multiple Choice

State if the triangles in each pair are similar. If so, state how you know they are similar.

• Yes, AA Similarity

• Yes, SSS Similarity

• Yes, SAS Similarity

• Not Similar

Proportion

An equation of two equivalent fractions

​

You can solve them by cross multiplying. ​

Some text here about the topic of discussion

Multiple Choice

Which equation would be the correct proportion to set up?

• $$\frac{8}{x}=\frac{72}{6}$$  

• $$\frac{x}{6}=\frac{72}{8}$$  

• $$\frac{x}{8}=\frac{72}{6}$$  

• $$\frac{6}{72}=\frac{x}{8}$$  

Corresponding Sides and Angles

• Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles. 

•  If two figures are similar, then the measures of the corresponding angles are equal and the ratios of the lengths of the corresponding sides are proportional.

Multiple Choice

Example 1, Part 1: 

A tree 24 feet tall casts a shadow 12 feet long. Brad is 6 feet tall. Which of the following is NOT a proportion that accurately represents the given example?

• $$\frac{24}{12}=\frac{6}{x}$$

• $$\frac{24}{6}=\frac{12}{x}$$

• $$\frac{24}{12}=\frac{x}{6}$$

Fill in the Blanks

Unit 5:
Right Triangle Trig

Multiple Choice

Solve for the missing angle.

• 50.5 degrees
• 12.7 degrees
• 36.9 degrees
• 72.2 degrees

Multiple Choice

Find x round to the nearest tenth.

• 46.1
• 4.9
• 73.5
• 68.6

Multiple Choice

Find the Tangent of <c

• 27/36
• 27/45
• 45/36
• 45/27

Multiple Choice

A yacht is anchored 90 feet offshore from the base of a lighthouse.  The angle of elevation from the boat to the top of the lighthouse is 26 degrees.  The distance between the yacht and the top of the lighthouse is about 100 feet.  Which of these is nearest to the height of the lighthouse? 

• 25 feet
• 45 feet 
• 110 feet 
• 135 feet 

Multiple Choice

A hawk sitting on a tree branch spots a mouse on the ground 15 feet from the base of the tree.  The hawk swoops down toward the mouse at an angle of 30 degrees. What is the distance from the tree branch to the mouse? 

• 7.5 ft 
• 15 ft 
• 26 ft 
• 30 ft 

Unit 6:
Circles

Arcs: Part of a Circle

Multiple Choice

Solve for a. outside(whole)=outside(whole)

• 4

• 6.93

• 8

• 16

Angle Relationships in Circles

• Vertex on the circle

• Vertex in the interior of the circle

• Vertex in the exterior of the circle.

Multiple Choice

Find the measure of arc AB.

• 17 degrees

• 34 degrees

• 68 degrees

• 146 degrees

Finding (h,k) and r

Multiple Choice

Find the center: $$\left(x+4\right)^2+\left(y-2\right)^2\ =\ 25$$

• (4, 2)

• (-4, -2)

• (-4, 2)

• (4, -2)

Multiple Choice

Find the radius: $$\left(x+4\right)^2+\left(y-2\right)^2\ =\ 25$$  

• 2.5

• 5

• 25

• 50

Multiple Choice

What is the standard equation of the outer boundary of the region serviced by the tower?

• (x + 3)² + (y - 5)² = 144

• (x - 3)² + (y + 5)² = 144

• (x + 3)² + (y - 5)² = 12

• (x - 3)² + (y + 5)² = 12

The triangle at the right is a right triangle, and θ is in standard position.

Match

Now, use the given triangle to represent $$\sin\theta$$ and $$\cos\theta$$ in terms of the variables given in the diagram.

• $$\frac{y}{1}=y$$

• $$\frac{x}{1}=x$$

• 1

sinθ

cosθ

OP

• sinθ

• cosθ

• OP

Multiple Choice

 Which point on the unit circle corresponds to  $$\frac{3\pi}{2}$$  ?

• (0, 1)

• (1, 0)

• (0, -1)

• (-1, 0)

Multiple Choice

$$\sin\ \left(\frac{7\pi}{2}\right)\ =\ ?$$  

Based on your unit circle,

• 1/2

• 0

• -1

• 1

Multiple Choice

What is the exact value of sin 150°

• -1/2

• -√3/2

• 1/2

• √3/2

Unit 7:
Equations and measurements

On this problem, we isolated for H, by multiplying by 3 to remove the 1/3 to start.

We then had to divide by (pi)r² to undo the multiplication on the right.

Multiple Choice

Bob has a rectangular prism gift box what it the volume of bobs gift box?

• 64in³
• 60in³
• 240in³
• 24in³

Multiple Choice

A cube has a surface area of 9/25 cm². What is it's volume?

• 3/5

• 18/50

• 27/125

• 9/25

Unit 8:
Probability & Stats

to find probability, you make a fraction.

top = number of times what you want to happen can happen.

bottom = total number of things that can happen.

since there are 12 marbles in the bag, 12 goes on the bottom.

in the first example, you want red. there are 7 red marbles so 7 goes on top.

in the 2nd examples you want blue. there are 5 blue marbles, so 5 is on top.

​

​

​

​

​

Multiple Choice

What is the intersection of sets A and B?

A ⋂ B = ______

• 1, 2, 3, 4. 5

• 1, 2

• 3, 4, 6, 8

• 3, 4

Multiple Choice

What is the Union of sets A and B?

A U B = ______

• 1, 2, 3, 4, 5, 6, 8

• 1, 2, 3, 4, 5

• 3, 4, 6, 8

• 3, 4

Independent & Dependent Events

• Independent - One activity does NOT effect the outcome of a different activity. (REPLACE)

• Dependent - One activity DOES effect the outcome of another activity. (do NOT replace)

Independent Events

• To find the probability that two independent events will happen - MULTIPLY the probabilities of the two events.

Find the probability of choosing a green marble at random from a bag containing 5 green and 10 white marbles and then flipping a coin and getting tails.

Dependent Events

You must determine the effect that the first event has on the probability of the second event.

A reading list contains 5 historical books and 3 science-fiction books. What is the probability that Juan will randomly choose a historical book for his first report and a science-fiction book for his second?

Multiple Choice

Find the probability of spinning an evenly divided spinner numbered 1-8 and getting a PRIME number on one spin and getting an ODD number on a second spin.

• $$\frac{1}{2}$$

• $$\frac{1}{4}$$

• $$\frac{5}{8}$$

• $$\frac{3}{16}$$

​A represents 1st event
B represents 2nd event (follows the word given)

​Intersection

When asked for "OR" (also "∪") situations, the addition rule is used.

• you must first decide if the situation is mutually exclus​ive or overlapping.

​Some text here about the topic of discussion.

Addition Rule​

​Mutually Exclusive Events

Events are mutually exclusive when they cannot happen together

​ For example:​

• Rolling a single​ die, you cannot roll a 3 and a 4 at the same time.

• Flipping a coin, you cannot have heads and tails at once.

• Pulling a card from a standard deck, you cannot pull a heart and spade together.​

​Overlapping Events

Events are Overlapping when they can happen together

​ For example:​

• Rolling a single​ die, you can roll a 3 and an odd number at the same time.

• Pulling a card from a standard deck, you can pull a heart and "4" together.​

Multiple Choice

A basket contains four apples, four

peaches, and four pears. You randomly

select a piece of fruit. It is an apple or a

peach.

• Mutually exclusive; [1 / 3 ~= 0.333]

• Not mutually exclusive; 8/11 = 0.727

• Not mutually exclusive; 7/12 = 0.583

• Mutually exclusive; 2/3 = 0.667

Multiple Choice

There are eleven shirts in your closet, four

blue and seven green. One of the blue

shirts and four of the green shirts fit well.

The others are too big. You randomly

select a shirt to wear. It is green or fits

well.

• Mutually exclusive; [1 / 3 ~= 0.333]

• Not mutually exclusive; 8/11 = 0.727

• Mutually exclusive; 7/12 = 0.583

• Not mutually exclusive; 2/3 = 0.667

A Permutation refers to a list of numbers in a definite order

Calculator

Directions​

​

1. Type in the n value

2. Press MATH

3. Move over to PRB​

4. Select nPr

5. Type in rthe r value

Multiple Choice

What does ₁₀P₅ mean?

• Permutations with 5 choices and 10 positions
• Permutations with 10 choices and 5 positions
• Permutations with 5 numbers and 10 operations
• Permutations with 5 hot dogs and 10 drinks

Multiple Choice

A team of 15 basketball players need to choose a Captain & a Co-captain. How many ways can that selection be made?

• 5140

• 280

• 210

• 35

​

1. ​Type in the value of n

2. Press MATH

3. Move over to PRB or PROB

4. ​Move down to nCr or press 3

5. ​Type in the value of r

6. Press enter​

​

Calculator

Multiple Choice

Ted and Julia are planning trips to six countries this year. There are 10 countries they would like to visit. They are deciding which countries to skip.

• 5140

• 280

• 210

• 35

Match

Decide whether you'd solve using a Permutation, Combination, or simply need a single factorial, then match the cards accordingly.

• Permutation

• Combination

• Factorial

A lock has a code of 3 numbers between 1 and 15. How many codes can be created if no numbers repeat

The principal would like to assemble a committee of 8 students from the 12-member student council.

Ribbons are being awarded for 1st through 5th place to the top 5 dogs in a dog show.

• A lock has a code of 3 numbers between 1 and 15. How many codes can be created if no numbers repeat

• The principal would like to assemble a committee of 8 students from the 12-member student council.

• Ribbons are being awarded for 1st through 5th place to the top 5 dogs in a dog show.

Multiple Choice

Find the number of possibilities:
A group of 25 people are going to run a race.  The top 8 finishers advance to finals.

• 625
• 4.36 X 10¹⁰
• 1,081,575

Multiple Choice

How many 2-digit numbers can you make using the digits 1, 2, 3, & 4 without repeating the digits?

• 90
• 100
• 12
• 24