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=Dilations=

OBJECTIVES:
~Using a scale factor, construct a dilation on a segment and a point. ~Dialate a closed plane figure.

VOCABULARY:
~Dilation: One figure made larger or smaller, but the new figure has the same angle measures. ~Scale Factor: In a dialation if you increase or decrease the origional figure by the scale factor, the two figures are preportional. Take the coordinates and multiply them by the scale factor. ~Contraction:The original figure is made smaller. ~Expansion: The original figure is made bigger. ~Preimage: The original figure ~Image:The image created due to an expansion or contraction. The above picture shows an example of a dilation. When you get your eyes dilated, your pupils get larger or expand just like a dilation.

EXAMPLES:
1) If the coordinates of a triangle are (1,6), (2,4) and (1,2), what are the coordinates of the new image if the scale factor is 2. 2) If the coordinates of a rectangle image are (2,4) (2,2) (6,2) and (6,4) and the preimage rectangle was (1,2) (1,1) (3,1) and (3,2), what is the scale factor? 3) What are the preimage coordinates of a triangle with the image coordinates of (6,9) (3,9) and (3,6) and a scale factor of 3?
 * answers at bottom*

OBJECTIVES:
~Define similar polygons. ~Use Properties of Proportions and scale factors to solve problems involving similar polygons.

VOCABULARY:
~Similar Figures: If you can dialate the figures to make them the same size then the figures are similar ~Polygon Similarity Postulate: The figures are similar if they meet these reqirements: the coinciding angles have the same angle maesure and the sides are preportional. ~Cross-Multiplication Property: If a/b=c/d and b and d does not 0, then ad equals bc ~Reciprocal Property: If a/b=c/d and a, b, c and d does not 0, then b/a equals c/d ~Exchange Property: If a/b=c/d and a, b, c and d does not 0, then a/c equals b/d ~"Add-One" Property: If a/b=c/d and b and d does not 0, then a+b/b equals c+d/d


 * This picture shows an example of similar figures. The giraffes are different sizes but everything else is the same.**

EXAMPLES:
1) If rectangle number 1 has side lengths of 6 and 14 while rectangle number 2 has side lengths of 3 and 7, can you prove that the rectangles are similar? 2) If rectangle number 1 has side lengths of 3 and 5 while rectangle number 2 has side lengths of 6 and 12, can you prove that the rectangles are similar? 3) If triangle number 1 has side lengths of 12, 8 and 4 while triangle number 2 has side lengths of 6, 4 and 3, can you prove that the triangles are similar?
 * answers at bottom*

OBJECTIVES:
~Use the AA Triangle Similarity Postulate and the SSS and SAS Triangle Similarity Theorems.

VOCABULARY:
~AA (Angle-Angle) Smilarity Postulate: When two given angles on the triangles have the same angle measure, then the triangles are similar. ~SSS (Side-Side-Side) Similarity Theorem: When all three sides of one trianle are preportional to the other triangles side, it means the triangles re similar. ~SAS (Side-Angle-Side) Similarity Theorem: When two sides and the angle between them are the same as the coinciding two sides and the angle in between them on the other triangle then the triangles are similar. This pictures shows an example of scale factor. This is a model airplane which is just a smaller version, or contraction, of a real airplane.

EXAMPLES:
1) If triangle number 1 has side lengths of 3, 4 and 5, and triangle number 2 has side lengths of 15, 20 and 25, are they similar? What postulate would prove if the triangles are similar? 2) If triangle number 1 has one angle measure of 63 and another angle measure of 46 and triangle number 2 has angle measures of 31.5 and 23, can these triangles be proven similar? With which Postulate? 3) If triangle number 1 has side lengths of 10 and 12 and triangle number 2 has side legths of 5 and 6, are they similar? Why or why not?
 * answers at bottom*

OBJECTIVES:
~learn to prove the Side-Splitting Theorem ~Solve problems using the Side-Splitting Theorem.

VOCABULARY:
~Side-Splitting Theorem: Two halves of a triangle have been split proportionally by a line which is parallel to one side of the triangle. ~Two-Transversal Proportionality Corollary: When two transversals are intersected by three or more parallel lines.

EXAMPLES:
1)The upper left hand side of the triangle is 24, the lower left hand side is 32, the upper right hand side 30, the lower right is x and you need to find x, so now you find x by cross multiplication. 2)The upper left is 14, the lower left is 28, upper right is x lower right is 30. Find x. 3)The upper left is 122, lower left is x, upper right is 34 lower right is 148. Find x.
 * answers at bottom*

OBJECTIVES:
~Measure distances inderectly using triangle similarity. ~For altitudes and medians of triangles use similarity theorems.

VOCABULARY:
~Proportional Altitudes Theorem: When the two triangles coinciding altitudes have the ratio of the coinciding sides they are similar. ~Proportional Medians Theorem: When the two triangles coinciding medians have the ratio of the coinciding sides they are similar. ~Proportional Angle Bisectors Theorem: When the two triangles coinciding angle bisectors have the ratio of the corresponding sides of the triangles. ~Proportional Segments Theorem: When the triangles angle bisectors divide the opposite side in half and those two segments have the ratio of the other two sides.

EXAMPLES:
1)Henry wants to find out the hight of the statue. Henry is 5 ft. tall his shadow is 4 ft long, the statue shadow length is 6 ft long, The hight of the statue is x. Think of the top of the statue being one side of a triangl, the shadow being the bottom of the triangle, and from the top of the statue to the end of the shadow being the third and do the same with henry, his shadow and from the top of his head to the end of his shadow. henry and his statue are both parallel to the groun and because of the position of the sun the angle of the top of the triangles and the bottoms are equal, so the triangles sre similar by the AA similsrity postulate. Now find the hight of the statue. 2) Imagine the situation is the same as in eqample one except its a flagpole and a buioding and the buildins hight is 40 ft tall, the buildings shadow is 34 ft. long, the flag poles' shaadow is x, and the flag pole is 15 ft. tall. Find the length of the flag poles shadow. 3)Triangle one is similar to triangle 2. PIcture in your head the right side(on triangle 1 it equals 4 and on triangle 2 it equals 7) is perpendicular to the bottom (on triangle 1 it equals 3 and on triangle 2 it equals 10) and the altitude on triangle two equals 5. Find the altitude of triangle 1.
 * answers at bottom*

OBJECTIVES:
~For areas of similar figures use ratios. ~Use ratios for similarsolids' volumes. ~Go through relationships between cross-sectional area, hight, and weight.

VOCABULARY:
NONE

EXAMPLES:
1)Square 1 has a a side length of 45, square 2 has a side length of 5. Knowing side of square1/ side of square2 is 45/5.Find the ratio for the area of the squares. 2)Circle1 has a radius of 100 and circle 2 has a radius of 10. Knowing radius of circle 1 over radius of circle 2 is 100/10. Find the ratio between the areas. 3)Cube1 has a side length of 57, and cube 2 has a side length of 31. Knowing edge of cube 1/ edge of cube2 is 57/31. Find the ratio of the volume of the cubes.
 * answers at bottom*

EXTRA HELP:
~[|Click here] for extra help on similar figures. To get to the help on similar figures, click on Similar Figures under the Relations and Size category. ~[|Click here] to view a movie that expalins similar triangles. Under the Video (Flash) Lessons, click on Similar Triangles. ~[|Click here] for extra help on similarity and proportions. Scroll down and click on Chapter 6: Similarity and Proportions. ~[|Click here] for extra help with all of the stuff talked about on this page. This page also contains games. Just click on the section you need more help with.

Section 1:
1) The coordinates are (2,12) (4,8) (2,4). 2) Scale Factor is 2. 3) The preimage coordinates are (2,3) (1,3) (1,2).

Section 2:
1) Yes, the rectangles are similar. 2) No, the rectangles are not similar because the scale factor is not the same. 3) No, the triangles are not similar because the scale factor is not the same.

Section 3:
1) The 2 triangles are similar and that can be proven by the SSS similarity theorem. 2) The 2 triangles are similar and that can be proven by the AA similarity postulate 3) No, the 2 triangles are not similar because there are only 2 side lengths and we don't know the angle measure between the sides so neither SSS or SAS would work.

Section 4:
1)To find x you put 24over 32 equals 30 over x. Then you cross multiply, so it would it be 24x equals32*30. 32*30 equals 960.Now divide that by 24. X=40. 2)To find x you put 14 over 28 then cross multiply that by x over 30. Now it's 14x is equal to 28*30.28*30 equals 840. Now divide that by 14. X=60. 3)To find x you put 122 over x is equal to 34 over 148.Cross multiply, so it's now 122*34 equals 148x.122*34 is 4148 now divide that by 148. X=28.02702703.

Section 5:
1) X over 5 equals 6 over 4. Cross multiply, so 4X equals 5*6, which is 30. Then divide that by 4 and your statue hight is 7.5 ft. tall. 2)40 over 15 equals 34 over x. Now Cross multiply, so 40X equals 15*34. 15*34 equals510 then divide that by 40. the flagpole is 12.75 ft. tall. 3)X/5 equals 3/10. times both sides by 5. so now X equals 3/10*5. The altitude of triangle 1 is 1.5.

Section 6:
1)6*6 equals 36. 2*2 equals 4. 36/4 reduces to 9/1. 2)Pi equals 3.14. 3.14*100 squared/ 3.14*10 squared. 3.14*10000/3.14*100. 31400/314 is the ratio of the area of the circles. 3)57*57*57is 185193. 31*31*31 is 29791. Volume of cube 1/ volume of cube2 is 185193/29791.