VOCABULARY – Write definitions for the math terms below ...
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Geometry
(MA/CP/ACC)
Summer Packet
2018-2019
Name: _____________________________
Supplies Needed For MA/CP/ACC Geometry:
1. 2” Binder and Loose Leaf Paper
2. Pencils
3. Folder
4. TI-83 or TI-84 Graphing Calculator
Objectives for Geometry Summer Packet
2018-2019
I. Finding the Equation of a Line ( Problems: #1- 8)
• Given a point that lies on that line and the y-intercept
• Given a point and a parallel line
• Graphing using the slope-intercept form
II. Solving Equations ( Problems: #9-14)
• Solving equations with variables on both sides
• Solving inequalities
III. Angle Pair Relationships ( Problems #15-17)
• Complementary Angles
• Supplementary Angles
• Linear Pairs
IV. The Slope Formula (Problems: #18-19)
• Use the counting method to find the slope of a line.
• Use the slope formula to find slope of a line.
V. Systems of Equations ( Problems: #21-23)
• Using the linear combination method to solve systems of equations
• Using the substitution method to solve systems of equations
VI. Solving Proportions ( Problems: #24-25)
• Solving proportions by cross multiplication
VII. Radicals ( Problems: #26-29)
• Simplifying radicals
• Rationalizing the denominator
VIII. Classifying Angles ( Problems: #30-33)
• Acute, right, obtuse or straight
IX. Binomials ( Problems: #34-37)
• Factoring polynomials
• Multiplying binomials (FOIL)
X. The Pythagorean Theorem ( Problems: #38-40 )
• Using the Pythagorean theorem to find missing lengths in right triangles
• Using properties of equality
XI. Classifying Polygons ( Problem: #41)
• Classifying polynomials
XII. Quadratic Equations (Problems: #42-43)
• Solving quadratic equations by taking the square root of both sides
• Using properties of equality
XIII. Distance Formula (Problems: #44-45)
• Use the distance formula to find the distance between two points
• Solving equations involving radicals
XIV. The Midpoint Formula (Problems: #46-77)
• Identifying the x coordinate and the y coordinate in an ordered pair
• Using the midpoint formula to find the midpoint of two points
ALGEBRA REVIEW
Finding the Equation of a Line
Example: Find an equation of the line, in slope intercept form, that passes through
the point (3, 4) and has a y-intercept of 5.
y = mx + b Write the slope-intercept form.
4 = 3m + 5 Substitute 5 for b, 3 for x, and 4 for y.
-1 = 3m Subtract 5 from each side.
[pic]= m Divide each side by 3.
The slope is [pic]. The equation of the line is [pic].
The slope of the parallel line is [pic].
Write the equation of the line, in slope intercept form, that passes through the given point and has the given y-intercept.
1. (2, 1); b = 5 _________________ 2. (7, 0); b = 13 ________________
3. (-2, -1); b = -5 ________________
4. Write the equation of a line that passes through (5, 1) and is parallel to
[pic]. (Hint: use the slope-intercept form to solve for b this time)
4. __________________
Graphing Linear equations
To graph a linear equation use the slope(m) and y intercept(b). First graph the b then count the slope up and over or down and over (if negative slope) (remember it has to be in slope-intercept form first- solve for y!!)
For example :
[pic], you would plot a point at [pic]3 on the y-axis then count up 2 and over to the right 3 units to plot another point, and connect the dots to make the line.
5. Graph [pic] 6. Graph [pic]
[pic] [pic]
7. Graph y = 2x − 1 8. Graph [pic]
[pic] [pic]
Solving Equations with Variables on Both Sides
Examples:
a. 6a – 12 = 5a + 9 b. 6(x + 4) + 12 = 5(x + 3) + 7
a – 12 = 9 Subtract 5a from each side. 6x + 24 + 12 = 5x + 15 + 7
a = 21 Add 12 to each side. 6x + 36 = 5x + 22
x = -14
Solve the equation.
9. 3x + 5 = 2x + 11 _______________ 10. 54c – 108 = 60c ____________
11. x + 6 = 9_____________ 12.4x + 2(x – 3) = 0 _______________ 2
Solving Inequalities
Examples:
Solve like an equation.
a. 3y + 1 > y − 3 b. 4x + 3 < 8x + 15
2y + 1 > −3 Subtract y from both sides −4x + 3 > 15 subtract 8x from both sides
2y > −4 Subtract 1 from both sides −4x > 12 subtract 3 from both sides
y > −2 Divide by 2 x < −3 divide by −4 (need to flip symbol)
13. 3x + 2 < 2x + 5 14. 4 − 5y [pic] 8 − y
13. _______________ 14. _______________
Angle Pair Relationships
Complementary Angles = 90° Supplementary Angles = 180°
[pic] [pic]
The measure of an angle is 46°.
15. The measure of the complement is _________. The measure of the supplement
is _________.
16. Two angles that form a linear pair lie on the same _______ and add to ______.
17. Two angles are supplementary. One angle measures 7x – 3 and the other
angle measures 12x – 7. Find the value of x and the measure of each angle.
x = _________ First Angle= __________ Second Angle= ________
****************************************************************************
Slope of a Line
Use the counting method to find the slope: Use the slope formula:
[pic]= [pic] [pic]
Count to find the slope
18.
[pic] 18. ______________________
Use the slope formula to find the slope of a line passing through the points given.
19. (3, 1) (5, 6) 20. (8, 10) (-3, -12)
19. ___________________ 20. ______________________
Solve the System of Equations
Example 1: Linear Combination Method
4 x – 3 y = -5
-4 x + 2 y = -16
The goal is to obtain coefficients that are opposites for one of the variables.
4 x – 3 y = -5
-4 x + 2 y = -16
-1 y = -21
-1 -1
y = 21
Substitute 21 for y: 4(21) – 3 y = -5. Solve to get y = -1. The solution is (21 , 89/3)
****************************************************************************
Example 2: Substitution Method
3 x + 2 y = 16
x + 3 y = 10 x = 10 – 3 y
Now substitute 10 – 3 y for x in the first equation: 3(10 – 3 y) + 2y = 16.
Solve for y to get y = 2.
Substitute 2 for y: x = 10 – 3(2). Solve to get x = 4. The solution is (4, 2).
****************************************************************************
21. 2 x – 3 y = -16 22. 7x + 2y = 10 23. y = 4x - 8
y = 5 x + 1 -7 x + y = -16 y = 2x + 10
Solving Proportions
Examples: a. [pic] Cross Multiply b. [pic] Cross Multiply
4x = 8 • 3 6 • 9 = x + 4
4x = 24 54 = x + 4
x = 6 50 = x
Solve for the variable.
24. [pic] _______________________ 25. [pic] _______________
Simplifying Radicals
Examples: a. [pic] b.[pic] c. [pic]
[pic] [pic]
[pic]
d. [pic] = [pic] e. [pic]
Simplify the expression.
26. [pic]= __________________ 27. [pic] = ___________________
28. [pic] = __________________ 29. [pic]= ________________
Classifying Angles
Classify the angle as acute, right, obtuse or straight.
30. 32° ____________ 31. 155° ____________
32. 180° ___________ 33. 90° _____________
Factoring and multiplying binomials
Examples: a. x2 – 5x - 14 b. (x + 3) (x - 5)
(x – 7) (x + 2) = x2 − 5x + 3x − 15 (FOIL)
x – 7 = 0 or x + 2 = 0 = x2 − 2x − 15 (Simplify)
x = 7 or x = -2
Factor each polynomial:
34. x2 + 5x - 24 _________________ 35. x2 – x - 20 _______________
Multiply each binomial:
36. (x + 8)(x − 9) = _______________ 37. (2x − 3)(x − 4) = ________________
Pythagorean Theorem
Examples: a. a = 12, b = 35, c = __________ b. a = 10, b = _______, c = 26
a2 + b2 = c2 a2 + b2 = c2
(12)2 + (35)2 = c2 (10)2 + b2 = (26)2
144 + 1225 = c2 100 + b2 = 676
1369 = c2 b2 = 576
[pic] = c b = [pic]
37 = c b = 24
c. A man leans a 8 ft. ladder against a house. The base of the
ladder is 2ft from the house. To the nearest tenth how high
on the house does the ladder reach?
Use Pythagorean’s theorem
8ft x2 + 22 = 82
x2 + 4 = 64
x2 = 60
2ft x [pic] 7.7 ft
Use the triangle above. Find the length of the missing side. Round answers to the nearest tenth. Remember: c is the hypotenuse.
38. a = 36, b = 15, c = _____________ 39. a = 17, b = _________, c = 49
40. A man leans a 12 ft. ladder against a house. The base of the
ladder is 4 ft. from the house. To the nearest tenth how high on the house does
the ladder reach?
40. _____________________
Classify Polygons
41. A polygon is named by the number of its sides. Fill in the chart below.
|Number of Sides |Type of Polygon |Number of Sides |Type of Polygon |
|3 | |8 | |
|4 | |9 | |
|5 | |10 | |
|6 | |11 | |
|7 | |12 | |
Solving Quadratic Equations
Example: x2 – 5 = 16
x2 = 21 Add 5 to both sides
x = [pic]
Exercises: Solve.
42. 4x2 + 5 = 45 ____________________43. 7x2 = 252 __________________
Distance Formula
Example: Find the distance between the points (-4, 3) and (-7, 8).
Formula:
Exercises: Find the distance between the points. Leave your answer as a simplified square root.
44. (-6, -6), (-3, -2) ______________ 45. (-8, 5), (-1, 1) _______________
Midpoint Formula
Example: Find the midpoint between (8, 14), (2, 6).
Formula:
The midpoint is always an ordered pair!
Exercises: Find the midpoint between the given points.
46. (-7, -17) and (11, 4) ________________________
47. (3, -8) and (-5, -13) ________________________
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