sculab5

=Chapter 8=

Dilations and Scale Factors
Dilation - an example of a transformation that's not rigid; it preserves the shape of a figure, but not necessarily the size You can find a dilation of a point by multiplying the x and y coordinates of a point by //n.// //D(x, y)// = (//nx//, //ny//) Scale factor - the number //n// Center of dilation - point on the same plane of a dilation that every line connecting a preimage point to an image passes through; every dilation has one Contraction - a dilation is a contraction if the size of a figure is reduced Expansion - a dilation is an expansion if the size of a figure is enlarged If |n| &lt; 1, the dilation is a contraction. If |n| &gt; 1, the dilation is an expansion.

Using the triangles above and you image that the sides are 5 and times that by 2 and the other side is 3 so the original triangle is 5, 3 and multiple that by two you get the second triangle 10, 6. The scale factor is 2 cause you take the image is 2 times bigger then the preimage.
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Good site explaining dilations with picture: http://www.regentsprep.org/Regents/math/geometry/GT3/Ldilate2.htm

Similar Polygons
Definition of a similar figure - 2 figures are similar if and only if one is congruent to the image of the other The symbol to show that figures are similar is ~. Polygon Similarity Postulate - 2 polygons are similar if and only if all of their angles are congruent and all of their sides are proportional

Their are two triangles one is 3, 8, 13 and the other is 4, 7,12 to find out if their similar you take 3/4 8/7 and 13/12 and see if their similar. 0.75, 1.14 and 1.08 now these triangels are not similar because the answers to the devision are not similar.
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Good site for triangle similarity: http://www.gcseguide.co.uk/similar_triangles.htm

__Properties of Proportions__ Cross-Multiplication Property If a/b= c/d and neither b or d= 0, then ad= bc. Reciprocal Property if a/b= c/d and none of the variables= 0, then b/a= d/c. Exchange Property If a/b= c/d and none of the variables= 0, then a/c= b/d. "Add One" Property If a/b= c/d and neither b or d= 0, then a+b/b= c+d/d.

Strategies for solving problems involving similar triangles: http://regentsprep.org/Regents/Math/similar/Lstrategy.htm

Triangle Similarity Postulates
Angle-Angle Similarity Postulate - If 2 angles are congruent to 2 angles of another triangle, then the triangles are similar. Side-Side-Side Similarity Theorem - If 3 sides of a triangle are proportional to 3 sides of another triangle, then the triangles are similar. Side-Angle-Side Similarity Theorem - If 2 sides of a triangle are proportional to 2 sides of another triangle and their included side is congruent, then the triangles are similar.

Draw 2 similar triangles, triangle ABC and triangle DEF. On triangle ABC, side AB= 10, side BC= 17, and side CA= 13. On triangle DEF, side EF= 34, and side FD= 26. What does side DE equal? Side BC corresponds to side EF. BC/EF= 17/34= 1/2. Side CA corresponds to side FD. CA/FD= 13/26= 1/2. So, we can use sides BC and EF or sides CA and FD to determine what side DE equals. BC/EF= 17/34= 1/2. Side AB corresponds to side DE. AB/DE= 10/X. For the sides to be proportional, which they are, in this case, X would have to have to be 2 times AB. 10 * 2= 20= DE Therefore, side DE equals 20 units.
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Powerpoint on the triangle similarity postulates we learned, with a sample problem: http://shs.smyrna.k12.de.us/users/amorabit/Geometry%20Powerpoints%5Cchapter08%5Clesson8_3.ppt Definition of similar polygons, an example, and cute cat pictures to help you get an idea of similarity: http://regentsprep.org/regents/math/similar/Lsimilar.htm

The Side-Splitting Theorem
[|Picture to use the Side-Splitting Therorem]

Side-Splitting Theorem - a line parallel to one side of the triangle divides the other 2 sides proportionally Two-Transversal Proportionality Corollary - 3 or more parallel lines divide 2 intersecting transversals proportionally

using the triangles above the 30 degree triangle is now 4/8 and the smaller triangle is now 2/X. You cross multiple and get 4X and 16. You then take 16 and divide that by the 4X and you get 4. 4/8= 2/X 16= 4X 16/4= 4X/4 4= X
 * Example:**

Site for the entire chapter, Side-Splitting Theorem, and a lot of great pictures: http://summit.k12.co.us/schools/Shs/StaffWebPages/YankowsK/IB4/IBMath4Ch8.htm

Indirect Measurement and Additional Similarity Theorems
Proportional Altitudes Theorem - if 2 triangles are similar, then their corresponding altitudes have the same ratio as their corresponding sides Proportional Medians Theorem - if 2 triangles are similar, then their corresponding medians have the same ratio as their corresponding sides Proportional Angle Bisectors Theorem - if 2 triangles are similar, then their corresponding angle bisectors have the same ratio as their corresponding sides Proportional Segments Theorem - an angle bisector of a triangle divides the opposite side into 2 segments that have the same ratio as their corresponding sides

You have two triangles (look at images above) the farthest right triangle is 8/10 and the other triangle is X/5. Now you have to find the value of X to see how the triangles are similar. So you cross multiple 8 times 5 which is equal to 40 and 10 times X which is 10X. You take 40 divided by 10X and get 4. The triangles are similar by 4.
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Area and Volume Ratios
[|Surface area ratio picture] [|Volume ratio picture]

An example for this section is the length of the prism is 12/8. Now find the volume. First you remember that the length is A/B and area is A^2/B^2, so the volume is A^3/B^3. So you take 12^3 or 12 times 12 times 12 to get 1728 then you take 8^3 or 8 times 8 times 8 to get 512. You then have 1728/512 is your volume. You divide these to numbers to get 3.375 is the ratio.
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Side with example for area ratio and volume ratio: http://www.tiem.utk.edu/~gross/bioed/bealsmodules/area_volume.html