Unit outline
sim 01-solve proportions using cross multiplication
To solve proportions, we use cross multiplication.
For example, with the first problem, we would cross multiply as shown in the bottom example. 4 * x = 4x 1 * 12 = 12 4x=12 Next, we would simply solve this problem like a normal equation. 4x/4=1 12/4=3 x=3 If we want to check our answer, we would just plug in x and cross multiply again. 4(3)=12 12(1)=12 12=12 If both sides of the equation are equal, then the proportion was solved correctly. Let's review with the second problem: -3(27)=81 -9(x)=9x -9x=81 -9x/9=x -81/9=9 - x=9 |
Sim 02-Perform transformations (translations, reflections, rotations, and dilations) in the coordinate plane with center at origin
Examples
Translation: A=(2,4) B=(4,4) C=(5,2) D=(2,1)
(x-7, y-3) A'=(-5,1) B'=(-3,1) C'=(-2,-1) D'=(-5,-2) Rotation: X=(1,2) Y=(3,5) Z=(-3,4)
*Rotation 180 degrees about the origin = (-x,-y) X'=(-1,-2) Y'=(-3,-5) Z'=(3,-4) |
Reflection: A=(-2,1) B=(2,4) C=(4,2)
*Flip over x-axis = (x, -y) A'=(-2,-1) B'(2,-4) C'=(4,-2) Dilation: A=(-2,-2) B=(-1,2) C=(2,1)
*enlargement- scale factor is 2 (ax, ay) A'=(-4,-4) B'=(-2,4) C'=(4,2) |
sim 03-identify scale factor, given two similar figures. also apply scale factor to find missing coordinates or side lengths/SIM 05-FIND MISSING SIDES OF TWO SIMILAR FIGURES USING PROPORTIONS
To identify scale factor, the formula is image/pre-image. If the absolute value of the scale factor (n) is greater than 1, then it is an enlargement, but if it is less than 1 it is a reduction. If it is equal to 1 then it is congruent.
This image to the left is an example of how you would find the missing coordinate or side length of a polygon.
One method using scale factor is to first find the scale factor which using the steps we found above would be 22/11 = 2 Now, we would apply to the other side by multiplying 6 by 2 and getting 12. The other method in solving it is to write a proportion. It would look something like this. 6/11=x/22 Then you would cross-mulitply and get 11x=132 Next you would divide by 11 and get your answer x=12 |
sim 04-verify & apply properties of proportions
The different properties that apply to all proportions are the Cross Multiplication Property, the Exchange Property of Proportions, the Reciprocal Property of Proportions, the Reciprocal Property of Proportions, and the Add One Property of Property of Proportions. Below shows the rule for each property, using variables.
SIM 06-DETERMINE IF POLYGONS ARE SIMILAR, THEN WRITE AN APPROPRIATE SIMILARITY STATEMENT, sim 07-determine if two triangles are similar using sss, sas, aa. then write an appropriate similarity statement
2 polygons are similar if
-all corresponding sides are proportional
-all corresponding angles are congruent
Just like with finding triangle congruence, we had shortcuts to determine congruence such as SSS, SAS, etc.
With similarity, it is the same concept except we only have 3.
They are:
SSS~ : If all corresponding sides are proportional then the two shapes are similar
AA~ : Of at least 2 angles in one triangle are congruent to 2 angles in another, then they are similar (because of corollary that if 2 angles are congruent then the third then has to be congruent, also)
SAS~ : If two sides are proportional, and one angle is congruent, then they are similar
-all corresponding sides are proportional
-all corresponding angles are congruent
Just like with finding triangle congruence, we had shortcuts to determine congruence such as SSS, SAS, etc.
With similarity, it is the same concept except we only have 3.
They are:
SSS~ : If all corresponding sides are proportional then the two shapes are similar
AA~ : Of at least 2 angles in one triangle are congruent to 2 angles in another, then they are similar (because of corollary that if 2 angles are congruent then the third then has to be congruent, also)
SAS~ : If two sides are proportional, and one angle is congruent, then they are similar
If were you given a set of two triangles like the one at right, you would say that these triangles are similar by AA~. This is because we know that vertical angles are congruent, and it already tells us that two angles are congruent. Though this triangle doesn't have any points an example of a similarity statement is triangle ABC ~ triangle DEF by AA~
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sim 50-construct a dilation giving the center of dilation with various positive and negative scale factors.
Here are two examples, one enlargement and one reduction, to the right that will show you how to perform different dilations
Enlargement 1. Draw a center of dilation labeled X 2. Draw 3 non-collinear points in space and connect them to create your first triangle, which will be your pre-image 3. Draw a line extending from X and connecting your first point labeled A 4. Use a compass to measure the distance from X to A and transfer that same point from A to create a new point called A' *Repeat steps 3 and 4 for the other two points 5. Connect your new points to create your image and write your similarity statement. Reduction 1. Draw a center of dilation labeled X along with 3 non-collinear points labeled A, B, and C 2. Connect them to create your first triangle, which will be your pre-image 3. Draw a line extending from X and connecting your first point labeled A 4. Because our reduction is n=1/2, make a perpendicular bisector to cut your line segment in half. This will create a new point which we will call A' *Repeat steps 3 and 4 for the other two points 5. Connect your new points to create your image and write your similarity statement. |
sim 08-prove that two triangles are similar using 2-column format
Now that we know what the statements are in proving similarity, we can use this information to create a proof. The example at left is one of the simplest versions. We are given that two angles in one triangle are congruent to two angles in another triangle. These would be our first two steps and the reason would simply be "given". We know that two congruent angles is enough to prove similarity so we can now write a triangle similarity statement with the reason AA~.
The example to the left is a little bit more complicated then the first one. Our given is actually AB/DB = BC/BE. This is a proportion and it shows that two of our sides are proportional to each other. Then we figure out that we have vertical angles, which in turn are congruent. We can now write a similarity statement by SAS~.
sim 09-use cpstp to determine the relationship between lengths in both proofs and algebra problems.
Similar to CPCTC, we would use CPSTP to show how different parts of a shape would be in proportion to each other. CPSTP stands for Corresponding Parts of Similar Triangles are Proportional.
For example, with the image to the right, we have a triangle with a mid segment. We are given that BD is parallel to AE and we are trying to prove that AC/BC = AE/BD, which essentially is showing that they are proportional. We would use what we have learned above about how to prove how two triangles are similar and then use CPSTP as our added step. You can see this as pictured in the image. |
sim 10-apply side-splitting theorem to find missing side lengths
The side-splitter theorem is essentially a more general case of the mid segment theorem. A visual depiction is shown in this Euler Diagram.
Essentially, the rule for the theorem states that if XY is parallel to BC, then a/b=c/d (*showing they are proportional)
Example: Replace a with 6, b with 2, c with 15, and d with x.
We would first start off by writing a proportion.
6/2 = 15/x
When we cross multiply, we get 6x=30
Divide it and get
x=5
Example: Replace a with 6, b with 2, c with 15, and d with x.
We would first start off by writing a proportion.
6/2 = 15/x
When we cross multiply, we get 6x=30
Divide it and get
x=5
Sim 11-find the ratio of areas for two similar figures, given scale factor. solve problems algebraically.
To understand this concept, let's use two squares for example. First, we are given one square with a length and width of 1 cm and a second square with a length and width of 2 cm. If we want to find the area of that square, we would simply multiply the length by the width, or square it. So the area of the first square would be 1x1 or 1cm squared, and the second square would be 2x2 or 4cm squared. Now, let's say we made this square into a cube. We would cube the first measurement. So, the volume would be 1x1x1 or 1cm cubed for the first cube and 2x2x2 or 8cm cubed for the second one.
Now, here are the formulas for length, area, and volume ratios.
Length Ratio (LR): a/b
Area Ratio (AR): a squared/ b squared
Volume Ratio (VR): a cubed/ b cubed
Also, for each problem, we have a set of steps to follow
Example: Let's say we are given a fish tank that is 5x the dimension of the original. If the original is 100 gallons, how much is the new one?
From this, we can determine that the scale factor is 5.
We can see that the length ratio is 5/1.
Following our formulas, we can determine that the area ratio is 5 squared/ 1 squared or 25/1
Then we can see that the volume ratio is 5 cubed/ 1 cubed or 125/1
We know that they are asking for volume so we would set up the proportion
25/1=x/100
When you solve that you would get that x = 12500
Now, here are the formulas for length, area, and volume ratios.
Length Ratio (LR): a/b
Area Ratio (AR): a squared/ b squared
Volume Ratio (VR): a cubed/ b cubed
Also, for each problem, we have a set of steps to follow
- Find which (LR, AR, VR) is given
- Find out all 3 - LR=___, AR=___, VR=___
- Which is being asked for? (LR, AR, VR)
- Set up proportion and solve
Example: Let's say we are given a fish tank that is 5x the dimension of the original. If the original is 100 gallons, how much is the new one?
From this, we can determine that the scale factor is 5.
We can see that the length ratio is 5/1.
Following our formulas, we can determine that the area ratio is 5 squared/ 1 squared or 25/1
Then we can see that the volume ratio is 5 cubed/ 1 cubed or 125/1
We know that they are asking for volume so we would set up the proportion
25/1=x/100
When you solve that you would get that x = 12500
sim 12-given 2 numbers, find geometric mean
The formula for geometric mean is:
If z is geometric mean of x, y then z=square root of (x times y)
For example, let's say we are given the numbers 2 and 32.
2 would represent x and 32 would represent y. Using that formula we would write:
z=the square root of (2 times 32)
z=the square root of (64)
z=8
If z is geometric mean of x, y then z=square root of (x times y)
For example, let's say we are given the numbers 2 and 32.
2 would represent x and 32 would represent y. Using that formula we would write:
z=the square root of (2 times 32)
z=the square root of (64)
z=8
sim 13-solve problems using right triangle similarity, formed by an altitude constructed in a right triangle.
From the image to right, we can determine 3 things:
-a is the geometric mean of x and c -b is the geometric mean of y and c -h is the geometric mean of x and y For example: if we knew that x was 9 and y was 4 then we could determine what h was by multiplying 9x4, which equals 36, and find the square root of that, which is 6. |