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GEOMETRY REVIEW PACKET “Obstacles are those frightful things you see when you take your eyes off your goal”
-Henry Ford
As of Spring 2016, geometry is no longer a prerequisite for MTH101 and MTH165
Spend time with the material in this packet and practice problems in MyMathTest. There are 8 modules in
MyMathTest that you should work through. There is a summary page in this packet for each of those modules.
You need to score at least a 46 on Compass Geometry to meet the geometry requirement and avoid taking
MTH070.
Below is information on when the Harper College Testing and Assessment Center is open. You need to make
sure you bring a photo ID. They will also give you a TI83 calculator that you are allowed to use on the exam.
NOTE: They are closed Wednesday – Saturday of Thanksgiving week.
GEOMETRY REVIEW PACKET (Geometry Review): Registration for January, 2017
REGISTERING:
1.
Go to the following website: www.mymathtest.com and click
3.
4.
Scroll down and click
Next to “Do you have a Pearson Education Account?” click “No” and then fill in a login name and password and then
re-enter your password. Write this information down someplace so that you won’t forget what you used.
Enter the following access code exactly as shown below:
5.
WMSLSS-­‐GIBLI-­‐LOACH-­‐PYRAN-­‐RUGBY-­‐VOTES 6.
7.
then click
Under
put your real name and an email address that you check on a regular basis. For the
school zipcode put 60067 and then pick WM RAINEY HARPER COLLEGE from the list of schools. Fill in the
security question information and then click
You can print the
page.
ACCESSING THE TESTS AND PRACTICE MATERIALS:
1. Go to the following website: www.mymathtest.com
2.
3.
Under
Under
type the login name and password that you just created and then click
then click
type the following course ID number:
XL1Z-­‐U10V-­‐701Z-­‐90W2
You should now see the following:
4.
On the announcement page, click
5.
Now you are ready to begin. Click
Prep for Compass Geometry Harper (2)
Click
so that the system will check if there are any plugins your computer needs.
. Click
(over on the right)
Click on the “+” next to
to expand the list of topics. Work through the checklist below.
o Read over the handout called “GEOMETRY REVIEW: R1.1: Lines and Angles” (These handouts are on the next pages)
o In MMT, click on “R1.1 Lines and Angles” and work through the problems. Use Help Me Solve This or View an Example if you need to.
o In MMT, click on
on the left side of the screen. Click on Practice Test R1.1. You can take this test multiple times for practice.
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Read over the handout called “GEOMETRY REVIEW: R1.2: Rectangles and Squares”
In MMT, click on “R1.2 Rectangles and Squares” and work through the problems. Use Help Me Solve This or View an Example if you
need to.
o
In MMT, click on
on the left side of the screen. Click on Practice Test R1.2. You can take this test multiple times for practice.
Read over the handout called “GEOMETRY REVIEW: R1.3: Parallelograms and Trapezoids”
In MMT, click on “R1.3 Parallelograms and Trapezoids” and work through the problems. Use Help Me Solve This or View an Example if
you need to.
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In MMT, click on
on the left side of the screen. Click on Practice Test R1.3. You can take this test multiple times for practice.
Read over the handout called “GEOMETRY REVIEW: R1.4: Triangles”
In MMT, click on “R1.4 Triangles” and work through the problems. Use Help Me Solve This or View an Example if you need to.
In MMT, click on
on the left side of the screen. Click on Practice Test R1.4. You can take this test multiple times for practice.
Read over the handout called “GEOMETRY REVIEW: R1.5: Circles”
In MMT, click on “R1.5 Circles” and work through the problems. Use Help Me Solve This or View an Example if you need to.
In MMT, click on
on the left side of the screen. Click on Practice Test R1.5. You can take this test multiple times for practice.
Read over the handout called “GEOMETRY REVIEW: R1.6: Volume and Surface Area”
In MMT, click on “R1.6 Volume and Surface Area” and work through the problems. Use Help Me Solve This or View an Example if you
need to.
In MMT, click on
on the left side of the screen. Click on Practice Test R1.6. You can take this test multiple times for practice.
Read over the handout called “GEOMETRY REVIEW: R1.7: Pythagorean Theorem”
In MMT, click on “R1.7 Pythagorean Theorem” and work through the problems. Use Help Me Solve This or View an Example if you
need to.
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o
In MMT, click on
on the left side of the screen. Click on Practice Test R1.7. You can take this test multiple times for practice.
Read over the handout called “GEOMETRY REVIEW: R1.8: Congruent and Similar Triangles”
In MMT, click on “R1.8 Congruent and Similar Triangles” and work through the problems. Use Help Me Solve This or View an Example
if you need to.
o
In MMT, click on
on the left side of the screen. Click on Practice Test R1.8. You can take this test multiple times for practice.
o
In MMT, click on
on the left side of the screen. Click on Practice Test Geometry. You can take this test multiple times.
6. There are 28 objectives to master. You can click on Study Plan, View Progress at any point to see how you are doing.
There is a link to another packet of geometry review materials on the Announcement page in MyMathTest
GEOMETRY REVIEW PACKET R1.1: Lines and Angles Objectives: 1. Use vocabulary to describe lines and angles 2. Identify and calculate measures of complementary and supplementary angles 3. Identify and calculate measures of congruent angles 4. Calculate measures of angles related to parallel lines Points, lines and rays A point is named with a capital letter. A line is named with any two points on the line in one of two ways: line AB or AB • parallel lines are lines that never intersect • perpendicular lines intersect at a 90° angle A ray is named with its endpoint first and then a second letter in the direction that extends indefinitely in one of two ways: ray AB or AB Kinds of angles An angle is formed when two rays come together at a common endpoint called a vertex. An angle is typically named with 3 letters that represents points: a point on one ray, the vertex and a point on the other ray. Sometimes an angle is named with a number in pictures with many angles. • An acute angle is an angle that measures between 0° and 90°. • A right angle measures 90° • An obtuse angle measures between 90° and 180° • A straight angle measures 180° • Complementary angles are two angles whose measures sum to 90° • Supplementary angles are two angles whose measures sum to 180° • Congruent angles are two angles with the same measure. In the figure below line a and b are parallel lines cut by another line c called a transversal. Several different kinds of congruent angles are formed when this happens. •
•
•
Vertical angles are angles opposite each other when two lines cross. The angles to the left and right of this “X” are vertical angles and are congruent. The angles in the top and bottom of this “X” are vertical angles and are congruent. In the picture above the following are all pairs of congruent vertical angles: angles 1 and 4, angles 2 and 3, angles 5 and 8, angles 6 and 7 Where the transversal cuts through each parallel line, 4 angles are formed. Angles in the same position relative to that intersection are congruent corresponding angles. In the picture above the following are all pairs of congruent corresponding angles: angles 1 and 5, angles 2 and 6, angles 3 and 7, angles 4 and 8 Alternate interior angles are formed when you make the letter “Z” forward or backwards where the top and bottom of the Z need to be parallel. In the picture above the following are pairs of congruent alternate interior angles: angles 3 and 6, angles 4 and 5 You are given a TI83 calculator to use when you take the Compass Geometry Placement Exam. GEOMETRY REVIEW PACKET R1.2: Rectangles and Squares Objectives: 1. Find the perimeter and area of rectangles and squares 2. Find the perimeter and area of composite shapes 3. Solve application problems To find the perimeter of a closed figure, you need to add up all the sides around its outer border. The perimeter is the distance around. In applications if material is being placed around the outer edge, it is a perimeter problem. Some common examples are fencing around a yard or framing around a picture. The area of a closed figure is the number of square units is takes to cover the figure. In applications if material is covering a surface, it is an area problem. Some common examples are amount of flooring, amount of sod, and amount of paint for walls. Perimeter and area of rectangles (a square is just a special rectangle where length = width) Assume each small square in the figure below measures 1 foot on each side. • Then the distance around the outer perimeter is 2 feet + 3 feet + 2 feet + 3 feet = 10 Feet • Area = the number of squares to cover the figure = 6 square feet. In this case we can count whole squares, but in general, the area of a rectangle can be computed as length times width. Composite shapes are shapes that are made up of other common shapes For example, to compute the area of the shape in In the figure below we want to find the area of the the picture below, you should be able to see how to yard. The lot is a rectangle and the house takes up break the shape into two rectangles and then a rectangle piece of the lot. To get what is left over compute the area of each rectangle and add those for the yard we need to compute the two rectangle areas together. areas and subtract them. In some applications you will first need to compute a perimeter or area (you must pick correct one) and use it to compute the cost of a project. • In the first figure above, we want to put up some fencing that costs $12 per meter. Fencing goes around the outside so we need perimeter. P = 11 m + 4 m + 9 m + 7 m + 2 m + 11 m = 44 m !! ! $"#
So now to get the cost, we have ! ∙ ! = $528 • In the first figure above, we want to put carpet in a room with that shape and the carpet costs $23/m2. Carpet covers the surface so we need area. The shape is made up of two rectangles so we add the area of each of those. Area = (4 m)(11 m) + (7 m)(2 m) = 44 m2 + 14 m2 = 58 m2 !" !!
$"#
So now to get the cost, we have ! ∙ !! = $1334 You are given a TI83 calculator to use when you take the Compass Geometry Placement Exam. GEOMETRY REVIEW PACKET R1.3: Parallelograms and Trapezoids Objectives: 1. Find the perimeter of parallelograms and trapezoids 2. Find the area of parallelograms and trapezoids 3. Solve application problems 4. Find the area of composite shapes To find the perimeter of a closed figure, you need to add up all the sides around its outer border. The perimeter is the distance around. In applications if material is being placed around the outer edge, it is a perimeter problem. Some common examples are fencing around a yard or framing around a picture. The area of a closed figure is the number of square units is takes to cover the figure. In applications if material is covering a surface, it is an area problem. Some common examples are amount of flooring, amount of sod, and amount of paint for walls. Parallelogram Trapezoid Area of a parallelogram: In the picture above, think about moving the triangle piece to the other side of the parallelogram. The two shapes together make a rectangle. The area of a rectangle is base times height where the base and height must be perpendicular to each other. Be careful not to use the length of the slanty side as the height of the parallelogram. Area of a parallelogram = (b)(h) Area of a trapezoid: In the picture above, think about putting together two trapezoids side by side where one is turned upside down. The two together make a parallelogram with a base that has length (B + b) and a height of h. So to get the area of the two trapezoids together you would take (B + b)(h). To get (! ! !)(!)
the area of just one of the trapezoids you would take ! . Be careful not to use the length of the slanty side as the height of the trapezoid. The base and height must always be perpendicular to each (! ! !)(!)
other. Area of a trapezoid = ! Composite shapes are shapes that are made up of other common shapes For example, to compute the area of the shape in In the figure below there is a parallelogram inside the picture below, you should be able to see how to of a trapezoid. If we want to find the area of the 4 break the shape into a rectangle and a corner areas we should be able to see that we need parallelogram and then compute the area of each to take the area of the trapezoid and subtract the shape and add those areas together. area of the parallelogram. (No numbers given to actually compute this) In some applications you will first need to compute a perimeter or area (you must pick correct one) and use it to compute the cost of a project. Do you want to put material around the edge (need to start with perimeter) or do you want to cover the surface (need to start with area)? You are given a TI83 calculator to use when you take the Compass Geometry Placement Exam. GEOMETRY REVIEW PACKET R1.4: Triangles Objectives: 1. Find the perimeter and area of a triangle and composite shapes 2. Find the measure of angles in a triangle 3. Solve application problems To find the perimeter of a closed figure, you need to add up all the sides around its outer border. The perimeter is the distance around. In applications if material is being placed around the outer edge, it is a perimeter problem. Some common examples are fencing around a yard or framing around a picture. The area of a closed figure is the number of square units is takes to cover the figure. In applications if material is covering a surface, it is an area problem. Some common examples are amount of flooring, amount of sod, and amount of paint for walls. Triangle Triangle inside of a Rectangle Area of a triangle: In the picture above, think about putting the triangle in a rectangle. The area of the triangle is half the area of the rectangle. The area of the rectangle is base times height, so the area of the triangle is half of the base times the height. The base and height must always be perpendicular to each (!)(!)
other. Area of a triangle = ! Composite shapes are shapes that are made up of other common shapes For example, to compute the area of the shape in In the figure below there is a triangle inside of a the picture below, you should be able to see how to rectangle. If we want to find the area of the shaded break the shape into a rectangle and a triangle and region, you would find the area of the rectangle and then compute the area of each shape and add those subtract the area of the triangle. areas together. In some applications you will first need to compute a perimeter or area (you must pick correct one) and use it to compute the cost of a project. Do you want to put material around the edge (need to start with perimeter) or do you want to cover the surface (need to start with area)? Triangle angle relations: • The sum of the 3 angles in a triangle is always 180°. When you see the little box symbol in the corner •
•
of an angle, , it means that the angle has a measure of 90° An isosceles triangle has 2 sides of equal length. The angles opposite those sides also have equal measure. An equilateral triangle has all 3 sides of equal length. All 3 angles have the same measure and since those measures must sum to 180°, each angle in an equilateral triangle has a measure of 60° You are given a TI83 calculator to use when you take the Compass Geometry Placement Exam. GEOMETRY REVIEW PACKET R1.5: Circles Objectives: 1. Find the radius and diameter of a circle 2. Find circumference and are of a circle 3. Find the area of composite shapes 4. Solve application problems The diameter is the distance across a circle at its widest point through its center. The radius is the distance from the center of the circle to the edge of the circle. • Given the diameter, the radius is the diameter divided by 2. • Given the radius, the diameter is 2 times the radius. Circle The circumference of a circle is the distance around the outer edge of the circle. In all of your spare time, take some circles (lids for example) and measure around them and divide by the length of the diameter. No matter what size circle you start with, this ratio comes out a little bit bigger than 3…always. In fact the number is 𝜋. The circumference of a circle is (𝜋)(𝑑). On the placement exam they may have you use 3.14 as an estimate for 𝜋. The area of a circle is found by taking 𝜋 times r2. Area of a circle = 𝜋𝑟 ! Composite shapes are shapes that are made up of other common shapes For example, to compute the area of the shape in In the figure below there is a small circle inside of a the picture below, you should be able to see how to larger circle. If we want to find the area of the break the shape into a rectangle and a circle and circular walkway around the pool, you would find then compute the area of each shape and add those the area of the larger circle and subtract the area of areas together. the smaller circle. In some applications you will first need to compute a circumference or area (you must pick correct one) and use it to compute the cost of a project. Do you want to put material around the edge (need to start with circumference) or do you want to cover the surface (need to start with area)? You are given a TI83 calculator to use when you take the Compass Geometry Placement Exam. GEOMETRY REVIEW PACKET R1.6: Volume and Surface Area Objectives: 1. Name solids and find the volume 2. Solve application problems involving volume 3. Find the volume and surface area of a cylinder or rectangular solid The volume of a 3-­‐dimensional object can be thought of as the number of cubes it takes to fill it. In application problems if you are asked how much a container holds you are being asked for its volume. A prism is a 3-­‐dimensional object that has two parallel congruent bases (top and bottom). Although the bases can be any polygon, we are only going to look at prisms with bases that are either rectangles or triangles. The faces on a prism are always rectangles (remember that a square is just a special rectangle). • To get the volume of a prism you need to take the area of its base (the base is either a rectangle or a triangle) times the height of the prism. Volume of prism = (area of base)(height of prism) • The surface area of a prism is the number of squares it takes to cover the outer surface of the 3-­‐
dimensional object. In the table below, underneath each prism there is a picture of what the outer covering of each prism looks like. Computing the surface area of the prism is just like computing the area of composite shapes in R1.2, R1.3 and R1.4. You need to add up the areas of all of the rectangular faces and then add to that the area of the 2 parallel bases. Rectangular Prism Triangular Prism Cube (special rectangular prism) A pyramid is a 3-­‐dimensional object that only has one base. We are only going to look at pyramids with bases that are either squares or triangles. The faces on a pyramid are always triangles. • To get the volume of a pyramid you need to take the area of its base (the base is either a square or a triangle) times the height of the pyramid and divide by 3. Volume of pyramid = (area of base)(height of pyramid)/3 • The surface area of a pyramid is the number of squares it takes to cover the outer surface of the 3-­‐dimensional object. In the table below, underneath each pyramid there is a picture of what the outer covering of each pyramid looks like. Computing the surface area of the pyramid is just like computing the area of composite shapes in R1.2, R1.3 and R1.4. You need to add up the areas of all of the triangular faces and then add to that the area of the one base. Triangular Pyramid Square Pyramid Cylinders, Cones and Spheres: Cylinder V = 𝜋𝑟 ! ℎ SA = 2𝜋𝑟 ! + 2𝜋𝑟ℎ Cone Sphere V = 𝜋𝑟 ℎ/3 V = ! 𝜋𝑟 !
!
!
You are given a TI83 calculator to use when you take the Compass Geometry Placement Exam. GEOMETRY REVIEW PACKET R1.7: Pythagorean Theorem Objectives: 1. Find square roots 2. Find unknown lengths using the Pythagorean Theorem 3. Solve application problems The square root of a number x is the number that you need to mupliply by itself to get the original given number. We use a symbol called a radical to designate that we want the square root of a number. Example: 36 = 6 because 6! = 36 Keep in mind that you have a calculator. To compute the square root of a number on the calculator, you need to push 2𝑛𝑑 𝑥 ! and then enter the number and then push 𝐸𝑁𝑇𝐸𝑅 You may be asked to compute a square root of a number and round the answer to a particular place value. It is important that you can round to the correct place value. Example: Compute 35 and round to the nearest tenth. • On the calculator you get 5.916079783. This number is between 5.9 and 6.0 on the number line. This number is closer to 5.9 than it is to 6.0, so 5.916079783 rounded to the nearest tenth is 5.9 Pythagorean Theorem: Given a right triangle like the one below, 𝑎! + 𝑏 ! = 𝑐 ! . The two sides of the triangle to form the right angle are called the legs of the triangle and the remaining side is called the hypotenuse. Right triangle: Finding length of hypotenuse: Finding length of leg: 𝑎! + 𝑏 ! = 𝑐 ! This is a right triangle, so 𝑎 ! + 𝑏 ! = 𝑐 ! (12)! + (16)! = 𝑐 ! 144 + 256 = 𝑐 ! 400 = 𝑐 ! 20 = c The length of the hypotenuse is 20 ft This is a right triangle, so 𝑎 ! + 𝑏 ! = 𝑐 ! 𝑎 ! + (72)! = (75)! 𝑎 ! + 5184 = 5625 𝑎 ! = 441 𝑎 = 21 The length of the other leg is 21 There are several application problems involving right triangles. Find the right triangle in the picture and use 𝑎! + 𝑏 ! = 𝑐 ! to get the missing side. You are given a TI83 calculator to use when you take the Compass Geometry Placement Exam. GEOMETRY REVIEW PACKET R1.8: Congruent and Similar Triangles Objectives: 1. Identify similar triangles 2. Write ratios for corresponding sides 3. Find unknown lengths in similar triangles 4. Solve application problems Congruent triangles are two triangles whose corresponding angles are equal in size and whose corresponding sides are equal in length. If you can mentally slide, flip and/or rotate one of the triangles such that it can be placed directly on top of the other triangle, then the triangles are congruent. There are several ways that you can prove that two triangles are congruent to each other. • SSS: If all three sides of one triangle have the same measures as the three sides of another triangle, then the triangles must be congruent. • SAS: If two sides and their included angle all have the same measures as two sides and an included angle of another triangle, then those two triangles must be congruent. • ASA: If two angles and their included side all have the same measures as two angles and their included side of another triangle, then those two triangles must be congruent. • AAS: If two triangles and a non-­‐included side all have the same measures as two angles and their non-­‐included side of another triangle, then those two triangles must be congruent. This should make sense because since the 3 angles sum to 180, the third angle in each triangle must also be the same measure and then you would have ASA again. Ø You cannot prove two triangles are congruent using SSA (this is the nice way of spelling ASS) In the picture below, the original triangle is triangle ABC. But the next picture shows two different triangles that can be constructed with the same two side lengths and non-­‐included angle. You can see that triangle PQR and triangle PQR’ can be made with the same two side lengths and non-­‐
included angle. Triangle PQR is congruent to the original triangle ABC, but triangle PQR’ is not. So you cannot prove triangles are congruent using SSA. (Don’t use ASS) Example: Determine which method can be used to prove the triangles are congruent. •
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Remember that the symbol means that the angle measures 90°. These triangles have two angles and an included side that all have the same measures as given by 39°, 23 mm, 90°. So we can use ASA. The other angle in each triangle must be 180° − 39° − 90° = 51° Since we said the triangles are congruent, the other corresponding legs must also have equal lengths and the hypotenuses must be the same length. GEOMETRY REVIEW PACKET R1.8: Congruent and Similar Triangles continued Many times when creating drawings on a computer we take objects and we either stretch them or shrink them by some scale factor. In this particular section we are going to look at stretching or shrinking just triangular-­‐shaped objects. The original triangle and the triangle that you get by stretching or shrinking it by some constant scale factor are called similar triangles. Two things happen when you stretch or shrink a triangle. • The 3 angles do not change size. If you start with a 30-­‐60-­‐90 triangle, you end up with a 30-­‐60-­‐90 triangle…the sides are just shorter or longer depending on whether you stretched it or shrunk it. • The corresponding sides of the two triangles have all been stretched (or shrunk) by a common scale factor. So if one side of the triangle is double in size, the other sides are also double what they used to be. Example: Find all of the ratios for the similar triangles. •
First, it is important to visually think about flipping and/or rotating one of the triangles so that the angles of the same size are in the same relative position in the triangles. Original picture Picture after flipping the second triangle Two triangles will have 3 corresponding sets of sides. If the triangles are similar, then the ratio of the lengths for each of these pairs must be the same (i.e. the scale factor is constant for all three) !"
!"
!"
o !" = ! To get a simplified fraction, type 3 0 ÷ 4 𝑀𝐴𝑇𝐻 𝐸𝑁𝑇𝐸𝑅 𝐸𝑁𝑇𝐸𝑅 to get ! •
o
!"
!"
!"
=
!"
!
!"
!"
To get a simplified fraction, type 6 0 ÷ 8 𝑀𝐴𝑇𝐻 𝐸𝑁𝑇𝐸𝑅 𝐸𝑁𝑇𝐸𝑅 to get ! !"
o !" = ! To get a simplified fraction, type 4 5 ÷ 6 𝑀𝐴𝑇𝐻 𝐸𝑁𝑇𝐸𝑅 𝐸𝑁𝑇𝐸𝑅 to get ! Ø Since this ratio is the same for all 3 pairs of corresponding sides, triangle ABC is similar to triangle PQR Congruent triangles Similar triangles Triangle ACB is congruent to triangle YZX by AAS Triangle RPQ is congruent to triangle VST by SAS NOTE: Congruent triangles are also similar triangles where the scale factor is 1. Triangle ABC is similar to triangle DEF These are similar triangles where the scale factor is 2. Triangle R is similar to triangle S These are similar triangles where the scale factor is 0.8 You are given a TI83 calculator to use when you take the Compass Geometry Placement Exam.