Download Honors Geometry Summer Assignment

Shaler Area
School District
Mathematics Department
Honors Geometry
Summer Assignment
Due: Thursday, September 3, 2015
*There will be a quiz on this material the second week of school.
Name: _________________________________________________
1
Honors Geometry Summer Assignment
The following topics will begin your study of Geometry. These topics are considered to be a review of your
previous math courses and will not be covered in length during the start of the school year.
Note: You should expect to purchase a scientific calculator for this course.
Section 1: Fractions
To Multiply Fractions:




Multiply the numerators of the fractions
Multiply the denominators of the fractions
Place the product of the numerators over the product of the denominators
Simplify the Fraction
Example: Multiply




2
9
and
3
12
Multiply the numerators 2(3)=6
Multiply the denominators 9(12)=108
6
Place the product of the numerators over the product of the denominators 108
6
1
Simplify the Fraction 108 = 18
To Divide Fractions:





Invert (i.e. turn over) the denominator fraction and multiply the fractions
Multiply the numerators of the fractions
Multiply the denominators of the fractions
Place the product of the numerators over the product of the denominators
Simplify the Fraction
2
3
Example: Divide 9 by 12





3
4
1) 12 ( )=
4)
20
3
4
=
2
3
2
12
Invert the denominator fraction and multiply 9 ÷ 12 = 9 ( 3 )
Multiply the numerators 2(12)=24
Multiply the denominators 9(3)=27
24
Place the product of the numerators over the product of the denominators 27
Simplify the Fraction
24
27
=
8
9
2)
1
5
5)
(
10
4
1
10
3
5
)=
=
2
3)
2
7
(
21
30
6)
2
5
8
10
=
)=
Section 2: Simplifying Algebraic Expressions
The difference between an expression and an equation is that an expression doesn’t have an equal sign.
Expressions can only be simplified, not solved. Simplifying an expression often involves combining like terms.
Terms are like if and only if they have the same variable and degree or if they are constants. Simplifying
expressions also refers to substituting values to get a resultant value of the expression.
Simplify the following expressions by combining like terms.
7) 3 + 2𝑦 2 − 7 − 5𝑥 − 4𝑦 3 + 6𝑥
8) 𝑥 2 + 𝑥 2 + 𝑥 + 𝑥
9) 4(3𝑥 − 2𝑥 3 + 5) − 6𝑥
10) 𝑥(2𝑥 − 3𝑥 4 + 2𝑦 − 5𝑥𝑦)
11) 8𝑎 − (7𝑏 − 4𝑎) − 3(4𝑎 + 2𝑏)
Evaluate the following expressions by substituting the given values for the variables.
12) 6𝑎2 − 2𝑏 + 4𝑎𝑏 − 5𝑎 ; 𝑎 = −3 𝑎𝑛𝑑 𝑏 = 4
13) – 𝑘 2 + 4𝑚 − 2𝑘𝑚 − (3𝑘 + 2𝑚); 𝑘 = −2 𝑎𝑛𝑑 𝑚 = 3
14) 3(4𝑐 − 2𝑑) + 𝑑(𝑑𝑐 2 + 7); 𝑐 = −2 𝑎𝑛𝑑 𝑑 = 3
Section 3: Solving Equations
When solving an equation, remember to combine like terms first. Terms are like if and only if they have the same
variable and degree or if they are constants. Then, take steps to isolate the variable by following the order of
operations backwards and doing the inverse operation.
Solve each equation and check your answer.
15) 3(4 – 3t) = -2
18)
2
𝑠
3
−5 = 4
16) 5 – 2(3t + 4) = -1
19)
1
(3𝑥
2
17) 2(3x – 4) = -3(x – 8)
+ 5) = −3
20)
3
1
(2𝑥
3
− 1) =
3
(𝑥
4
+ 2)
Solve each equation for the indicated variable.
1
bh , solve for h
2
21) A  bh , solve for h
22) A 
23) 𝑝 = 2(𝑙 + 𝑤), solve for 𝑙
24) 𝐴 = 𝜋𝑟 2 , solve for 𝑟
5
25) 7 = 𝑥 , solve for x
Section 4: Graphing Linear Equations
A linear function is a function where the highest power of x is 1. You have seen these functions in many forms.
Some of the common forms are y = mx + b (slope-intercept form) and Ax + By = C (standard form). Notice in both
forms that the exponent for x is 1.
Every linear function has an x and y intercept.


x – intercept: Where a function crosses the x – axis. The coordinate is represented by (x, 0).
y – intercept: Where a function crosses the y – axis. The coordinate is represented by (0, y).
A key concept to consider when thinking of linear functions is slope.
Slope is the “m” in the y = mx + b and is defined to be –A/B for standard form of a line. Here are some definitions
of slope:
𝑠𝑙𝑜𝑝𝑒 = 𝑚 =
Positive slopes increase from left to right.
𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒
𝑟𝑖𝑠𝑒 𝑦2 − 𝑦1
=
=
ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 𝑟𝑢𝑛 𝑥2 − 𝑥1
Negative slopes decrease from left to right.
positive slope
negative slope
4
If a line has a slope of zero, it is horizontal and the equation is y = b
If a line has an undefined slope, it is vertical and the equation is x = c
Parallel lines have the same slopes.
Perpendicular lines have slopes that are opposite reciprocals. (opposite-different sign; reciprocal-flip fraction)
Formulas for equations of a line:
Slope-Intercept: y = mx + b
Point-Slope Form: y – y1=m(x – x1)
5
Graph and label each linear equation. (Note: You may need to put the equation in slope-intercept form.)
Are the lines parallel, perpendicular or neither? Can you tell by the equation?
26) 𝑦 = 3𝑥 − 2
6𝑥 − 2𝑦 = −4
27) 𝑦 =
−1
𝑥
3
+4
𝑦 = 3𝑥 − 1
______________________
______________________
______________________
______________________
28) 𝑦 – 2𝑥 = −5
1
𝑦 = 2𝑥
29) 𝑦 + 3 =
−3
𝑥
4
𝑦– 2 =
______________________
______________________
______________________
______________________
6
1
𝑥
3
Section 5 - Factoring Numbers
"Factors" are the numbers you multiply to get another number. For instance, the factors of 15 are 3 and 5,
because 3×5 = 15. Some numbers have more than one factorization (more than one way of being factored). For
instance, 12 can be factored as 1×12, 2×6, or 3×4. A number that can only be factored as 1 times itself is called
"prime". The first few primes are 2, 3, 5, 7, 11, and 13. The number 1 is not regarded as a prime, and is usually not
included in factorizations, because 1 goes into everything. (The number 1 is a bit boring in this context, so it gets
ignored.)
You most often want to find the "prime factorization" of a number: the list of all the prime-number factors of a
given number. The prime factorization does not include 1, but does include every copy of every prime factor. For
instance, the prime factorization of 8 is 2×2×2, not just "2". Yes, 2 is the only factor, but you need three copies of
it to multiply together to get 8, so the prime factorization includes all three copies.
On the other hand, the prime factorization includes ONLY the prime factors, not any products of those factors. For
instance, even though 2(2) = 4, and even though 4 is a divisor of 8, 4 is NOT in the PRIME factorization of 8. That is
because 8 does NOT equal 2(2)(2)(4)! This accidental over-duplication of factors is another reason why the prime
factorization is often best: it avoids counting any factor too many times. Suppose that you need to find the prime
factorization of 24. Sometimes a student will just list all the divisors of 24: 1, 2, 3, 4, 6, 8, 12, and 24. Then the
student will do something like make the product of all these divisors: 1(2)(3)(4)0(6)(8)(12)(24). But this equals
331776, not 24. So it's best to stick to the prime factorization, even if the problem doesn't require it, in order to
avoid either omitting a factor or else over-duplicating one.
In the case of 24, you can find the prime factorization by taking 24 and dividing it by the smallest prime number
that goes into 24: 24 ÷ 2 = 12. (Actually, the "smallest" part is not as important as the "prime" part; the "smallest"
part is mostly to make your work easier, because dividing by smaller numbers is simpler.) Now divide out the
smallest number that goes into 12: 12 ÷ 2 = 6. Now divide out the smallest number that goes into 6: 6 ÷ 2 = 3.
Since 3 is prime, you're done factoring, and the prime factorization is 2(2)(2)(3).
Find the prime factorization of 1050.
The above problem could be worked out like this:
The answer would then be 2(3)(5)(5)(7) or 2(3)(5)2 (7).
30) Find the prime factorization of 500.
31) Find the prime factorization of 1092.
7
There are some divisibility rules that can help you find the numbers to divide by. If the number is even, then it's
divisible by 2.


If the number's digits sum to a number that's divisible by 3, then the number itself is divisible by 3.
If the number ends with a 0 or a 5, then it's divisible by 5.
Of course, if the number is divisible twice by 2, then it's divisible by 4; if it's divisible by 2 and by 3, then it's
divisible by 6; and if it's divisible twice by 3 (or if the sum of the digits is divisible by 9), then it's divisible by 9. But
since you're finding the prime factorization, you don't really worry about these non-prime divisibility rules.
If you run out of small primes and you're not done factoring, then keep trying bigger and bigger primes (11, 13,
17, 19, 23, etc) until you find something that works — or until you reach primes whose squares are bigger than
what you're dividing into. Why? If your prime doesn't divide in, then the only potential divisors are bigger
primes. Since the square of your prime is bigger than the number, then a bigger prime must have as its remainder
a smaller number than your prime. The only smaller number left, since all the smaller primes have been
eliminated, is 1. So the number left must be prime, and you're done.
Section 6 - Square Roots
"Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo" a power or exponent
with a radical. The
symbol is called the "radical" symbol. The expression √9 is read as "the square root of
nine". If you square 3, you get 9 and if you square -3, you get 9. C
32 = 9 and (−3)2 = 9, so √9 = ±3
Simplifying Square-Root Terms
To simplify a square root, you "take out" anything that is a "perfect square"; that is, you take out front anything
that has two copies of the same factor:
The value of the “principal root” is positive. Both +2 and –2 squared give you 4, so both are roots and represented
in the form ±2. Sometimes a radical is not a perfect square, but it may "contain" a square amongst its factors. To
simplify, you need to factor the number under the radical and "take out" anything that is a square; you find
anything you've got a pair of inside the radical, and you move it out front. In other words, radicals can be
manipulated similarly to powers:
8

Simplify √144
There are various ways you can approach this simplification. One would be by factoring and then taking
two different square
roots:
The principal square root of 144 is 12.
You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. But my steps above
show how you can switch back and forth between the different formats (multiplication inside one radical, versus
multiplication of two radicals) to help in the simplification process.

Simplify √24 √6
Neither of 24 and 6 is a perfect square, but what happens if you multiply them inside one radical?

Simplify √75
You don't have to factor the radicand all the way down to prime numbers when simplifying. As soon as you see a
pair of factors or a perfect square, you've gone far enough.

Simplify √72
Since 72 factors as 2×36, and since 36 is a perfect square, then:
Since there had been only one copy of the factor 2 in the factorization 2×6×6, the left-over 2 couldn't come out of
the radical and had to be left behind.

Simplify √4500
Multiplying Square Roots
The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots. We use the fact
that the product of two radicals is the same as the radical of the product, and vice versa.

Write as the product of two radicals: √6
Co992011 All Rights Reserved
9

Simplify by writing with no more than one radical: √2 √8

Simplify by writing with no more than one radical: √3 √6
Adding (and Subtracting) Square Roots
Just as with "regular" numbers, square roots can be added together. But you might not be able to simplify the
addition all the way down to one number. Just as "you can't add apples and oranges", you cannot combine
"unlike" radicals. To add radical terms together, they have to have the same radical part.

Simplify: 2√3 + 3√3
Since the radical is the same in each term (namely, the square root of three), you can combine the terms.
You will have two copies of the radical, added to another three copies. This gives you five copies:
The middle step, with the parentheses, shows the reasoning that justifies the final answer.

Simplify: √3 + 4√3
The radical part is the same in each term, so you can do this addition. To help you keep track that the first
term means "one copy of the square root of three", insert the "understood" "1":
Don't assume that expressions with unlike radicals cannot be simplified. It is possible that, after simplifying the
radicals, the expression can indeed be simplified.

Simplify: √9 + √25
To simplify a radical addition, you must first see if you can simplify each radical term. In this particular
case, the square roots simplify "completely" (that is, down to whole numbers):
10

Simplify: 3√4 + 2√4
There are three copies of the radical, plus another two copies, but you can simplify those radicals right
down to whole numbers:
You might not see a simplification right away. If you hadn't noticed until the end that the radical simplified, your
steps would have been different, but your final answer would have been the same:

Simplify: 3√3 + 2√5 + √3
You can only combine the "like" radicals, so you’ll end up with two terms in your answer:
The expression
32) Simplify: 3√8 + 5√2
would also be an acceptable answer.
Copyright © Elizabeth Stapel1999-2011 All Rights Reserved
33) Simplify: √18 − 2√27 + 3√3 − 6√8
34) Simplify: 2√3 + 3√5
These two terms have "unlike" radical parts, and you can't take anything out of either radical. Since you can't
simplify the expression 2√3 + 3√5 any further and the answer is: 2√3 + 3√5

Expand: √2 (3 + √3)
To expand (that is, to multiply out and simplify) this expression, you first need to take the square root of
two and distribute through the parentheses:
11
35) Expand: √3 (2√3 + √5)
36) Expand: (1 + √2)(3 − √2)
How to Rationalize a Denominator
Radicals are never permitted to remain in the denominator of a fraction. When you have a single square root in
the denominator you multiply the numerator and denominator by that value.
For the following problems, the instruction is to rationalize the denominator, which means to write an equivalent
expression for it that doesn't have any radicals in the denominator.
Example 1:
Solution:
Explanation: To eliminate the radical in the denominator, multiply the numerator and denominator by the
radical (which is equivalent to multiplying by 1). The square root of 3 times the square root of 3 is 3. By
multiplying the numerator and the denominator by the square root of 3, we eliminate the square root or radical in
the denominator.
Example 2:
Solution:
Explanation: First, split up the quotient into two separate radicals. Then, just like the previous example, multiply
the numerator and denominator by the square root that is in the denominator, which in this case is the square
root of 6.
37)
8
√7
38)
√180
√9
39)
5
√2
12
40)
4
√20
Section 7 - Simple Polynomial Factoring
Factoring polynomial expressions is not quite the same as factoring numbers, but the concept is very similar.
When factoring numbers or factoring polynomials, you are finding numbers or polynomials that divide out evenly
from the original numbers or polynomials. But in the case of polynomials, you are dividing numbers and variables
out of expressions, not just dividing numbers out of numbers.
Previously, you have simplified expressions by distributing through parentheses, such as:
2(x + 3) = 2(x) + 2(3) = 2x + 6
Simple factoring in the context of polynomial expressions is backwards from distributing. That is, instead of
multiplying something through the parentheses, you will be seeing what you can take back out and put in front of
a parentheses, such as:
2x + 6 = 2(x) + 2(3) = 2(x + 3)
The trick is to see what can be factored out of every term in the expression. Remember that "factoring" means
"dividing out and putting in front of the parentheses". Nothing "disappears" when you factor; things merely get
rearranged.

Factor 3x – 12
The only thing common between the two terms (that is, the only thing that can be divided out of each
term and then moved up front) is a "3". So factor this number out to the front:
3x – 12 = 3(
)
When you divided the "3" out of the "3x", you were left with only the "x" remaining. Put that "x" as your
first term inside the parentheses:
3x – 12 = 3(x
)
When you divided the "3" out of the "–12", you left a "–4" behind, so put that in the parentheses, too:
3x – 12 = 3(x – 4)
This is the final answer: 3(x – 4)
*(Be careful not to drop "minus" signs when you factor.)
You can also apply the concept of the Greatest Common Factor, or GCF. In that case, you would methodically find
the GCF of all the terms in the expression, put this in front of the parentheses, and then divide each term by the
GCF and put the resulting expression inside the parentheses. The result will be the same.
41) Factor 7x – 7
42) Factor 12y2 – 5y
43) Factor x2y3 + xy
44) Factor 3x3 + 6x2 – 15x
13
Section 8 - Multiplying Polynomials
FOIL Method
The FOIL method is a special case of a more general method for multiplying algebraic expressions using the
distributive law.




First (“first” terms of each binomial are multiplied together)
Outer (“outside” terms are multiplied—that is, the first term of the first binomial and the second term of
the second)
Inner (“inside” terms are multiplied—second term of the first binomial and first term of the second)
Last (“last” terms of each binomial are multiplied)
The general form is:
Once you have multiplied by using the FOIL method, you must combine any like terms.

Use FOIL to simplify (x + 3)(x + 2)
"first": (x)(x) = x2
"outer": (x)(2) = 2x
"inner": (3)(x) = 3x
"last": (3)(2) = 6
So: (x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
45) Multiply (x + 5) (x + 7)
46) Multiply (y – 3) (y – 5)
47) Multiply (4x + 2) (4x – 2)
48) Multiply (2a + 3) (3a – 4)
14
Distribution Method
Sometimes you will have to multiply one multi-term polynomial by another multi-term polynomial. This type of
multiplication will use a form of distribution for one of the polynomials to the other polynomial.

Simplify (x + 3)(4x2 – 4x – 7)
(x + 3) (4x2 – 4x – 7)
= (x)(4x2 – 4x – 7) + (3)(4x2 – 4x – 7)
= 4x2(x) – 4x(x) – 7(x) + 4x2(3) – 4x(3) – 7(3)
= 4x3 – 4x2 – 7x + 12x2 – 12x – 21
= 4x3 – 4x2 + 12x2 – 7x – 12x – 21
= 4x3 + 8x2 – 19x – 21
49) Multiply (x – 2) (x2 + 4x + 4)
50) Multiply (2x + 7) (2x2 + 5x – 4)
Section 9 - Formula Review
There are many geometric formulas that we will explore over the course of the school year. These formulas will
relate to dimensions such as height, width, length, or radius to find perimeter, area, surface area, volume, etc.
There are some formulas that you have worked with in previous math courses and your instructor will expect you
to know them.
For instance, to find the area A of a rectangle: multiply the length l times the width w:
Arect = lw Copyright © Elizabeth
When looking at a picture of a rectangle, remember that "perimeter" means "length around the outside", you'll
see that a rectangle's perimeter P is the sum of the top and bottom lengths l and the left and right widths w:
Prect = 2l + 2w or 2(l+w)
15
Because the length and width of a square is identical, there are special formulas that we can use for area and
perimeter. The area A and perimeter P of a square with side-length s are given by:
Asqr = s2
Psqr = 4s
You should know the formula for the area of a triangle. Given the measurements for the base b and the height h
of the triangle, the area A is:
Atri = (1/2)bh
The perimeter P of the triangle is the sum of the lengths of the triangle's three sides.
You should know the formula for the circumference C and area A of a circle, given the radius r:
Acir = (𝜋)r2
Ccir = 2(𝜋)r or (𝜋)d
("pi" (𝜋) is the number approximated by 3.14159 or the fraction
22
7
Remember that the radius of a circle is the distance from the center to the outside of a circle. In other words, the
radius is just halfway across. If you are given the diameter, the length of a segment going all the way across the
circle through the center, then you'll first have to divide it in half in order to apply some of the above formulas.
51) Find the perimeter and the area of the given quadrilaterals.
36 in.
47 ft
40 in.
33 in.
57 ft
52) The figure is formed from rectangles. Find the perimeter and total area of the figure. The diagram is not to
scale.
2 ft
8 ft
2 ft
10 ft
16
53) Find the area of the given triangle.
5 cm
6.4 cm
54) Find the circumference and area of the given circle to the nearest tenth.
39 in.
Thank you for taking the time to complete this assignment.
Your Geometry class will begin with these questions next year.
Have a great summer!!!
17