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Similarity Two geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, or one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all parabolas are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. If two angles of a triangle are equal to two angles of another triangle, then the triangles are similar. What are similar polygons? Similar polygons are polygons for which all corresponding angles are congruent and all corresponding sides are proportional. Example: Many times you will be asked to find the measures of angles and sides of figures. Similar polygons can help you out. 1. Problem: Find the value of x, y, and the measure of angle P. Solution: To find the value of x and y, write proportions involving corresponding sides. Then use cross products to solve. 4 x -=6 9 6x = 36 x=6 4 7 -=6 y 4y = 42 y = 10.5 To find angle P, note that angle P and angle S are corresponding angles. By definition of similar polygons, angle P = angle S = 86o. Special Similarity Rules for Triangles The triangle, geometry's pet shape, has a couple of special rules dealing with similarity. They are outlined below. 1. Angle-Angle Similarity - If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. 1. Problem: Prove triangle ABE is similar to triangle CDE. Solution: Angle A and angle C are congruent (this information is given in the figure). Angle AEB and angle CED are congruent because vertical angles are congruent. Triangle ABE and triangle CDE are similar by Angle-Angle. 2. Side-Side-Side Similarity - If all pairs of corresponding sides of two triangles are proportional, then the triangles are similar. 3. Side-Angle-Side Similarity - If one angle of a triangle is congruent to one angle of another triangle and the sides that include those angles are proportional, then the two triangles are similar. 2. Problem: Are the triangles shown in the figure similar? Solution: Find the ratios of the corresponding sides. UV 9 3 -- = -- = KL 12 4 VW 15 3 -- = -- = LM 20 4 The sides that include angle V and angle L are proportional. Angle V and angle L are congruent (the information is given in the figure). Triangle UVS and triangle KLM are similar by Side-Angle-Side. Quiz on Similarity of Triangles 1. All congruent triangles are similar, but all similar triangles are not congruent. True False 2. In a triangle ABC, a line is drawn parallel to BC, which cuts AB at D and AC at E. The ratio of AD:AB is equal to 3:5. What is the ratio of DE:BC? 3:4 Impossible to say 3:5 4:5 3. What is the sign which denotes similarity? ~ := ^ --- 4. In a triangle XYZ, a line 'PQ' is drawn parallel to YZ, cutting XY at P and XZ at Q. The ratio of XP:XY is 2:3. What is the ratio XQ:QZ? 2:3 2:1 Impossible to say 2:5 5. If a line divides two sides of a triangle in proportion, then it is parallel to the third side. True False 6. The Midpoint Theorem states: "A line joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half the length of the third side." So, riddle me this: What is the converse of the Midpoint Theorem? None of these are the converse. "If a line is parallel to one side of a triangle and is equal to half the length of the same side, the given line passes through the midpoints of the other two sides." "If a line passes through the midpoint of one side of a triangle and is parallel to a second side, the given line bisects the third side." "If a line passes through the midpoint of one side of a triangle, and is equal to half the length of a second side, it bisects the third side." 7. Enough about sides. Let's talk areas. You have been given two similar triangles, MNO and DEF. You have to find out the ratios of their areas. Which of the following formulae can be used to find out the ratio of the areas of the given triangles. The ratio of the areas of the triangles is equal to the square of the corresponding sides The ratio of the areas of the triangles is equal to the square of the corresponding altitudes The ratio of the areas of the triangles is equal to the square of the corresponding medians All of these can be used to evaluate the ratio of the areas of the triangles 8. All equilateral triangles are similar. True False 9. A given triangle ABC is isosceles. Angle B and angle C are the base angles. From B, a perpendicular 'BD' is drawn to cut AC at point D. From C, a perpendicular 'CE' is drawn to cut AB at point E. State which of the following pairs of triangles are similar. DBC ~ AEC BDC ~ AEB BDC ~ CEB CDB ~ BCA 10. ABC and PQR are two triangles. If the ratios of all the corresponding sides of the triangles are equal, then ABC ~ PQR. True False Answers to the Quiz 1. The correct answer was true To define similarity simply, it is a case in which all the corresponding angles of two or more triangles are equal, but their sides are not necessarily equal. Congruent triangles are triangles which are identical in every way. As such, all congruent triangles are similar, but all similar triangles are not congruent. 2. The correct answer was 3:5 Triangle ADE was similar to triangle ABC. A striking property of similar triangles is that the ratio of all their corresponding sides are equal. For example, if triangle ABC is similar to triangle PQR, then :AB:PQ=AC:PR=BC:QR 3. The correct answer was ~ ABC~PQR means that triangle ABC is similar to riangle PQR. 4. The correct answer was 2:1 In order to evaluate this answer, you had to use the basic proportionality theorem, which states, “if a line is drawn parallel to a side of a triangle, it divides the other two sides in proportion.” For example, in the above mentioned case, XP:PY-XQ:QZ. Therefore, since the ratio XP:XY was given, you were required to find out the ratio of XP:PY. XP:XY= 2:3 Let XP=2x and XY be 3x PY=XY-YP= 3x-2x=x Therefore XP:PY=2x:x=2:1 But XP:PY=XQ:QZ Therefore, XQ:QZ=2:1 5. The correct answer was true In ABC, DE is a line which cuts AB at D and AC at E. AD:DB=AE:EC. Therefore, DE is parallel to the third side of ABC, namely BC 6. The correct answer was “if a line passes through the midpoint of one side of a triangle and is parallel to a second side, the given line bisects the third side.” 7. The correct answer was the ratio of the area of the triangles is equal to the square of the corresponding sides 8. The correct answer was true. 9. The correct answer was BDC~CEB 10. The correct answer was true. Parallel lines & Triangles What do parallel lines and triangles have to do with similar polygons? Well, you can create similar triangles by drawing a segment parallel to one side of a triangle in the triangle. This is useful when you have to find the value of a triangle's side (or, in a really scary case, only part of the value of a side). The theorem that lets us do that says if a segment is parallel to one side of a triangle and intersects the other sides in two points, then the triangle formed is similar to the original triangle. Also, when you put a parallel line in a triangle, as the theorem above describes, the sides are divided proportionally. 1. Problem: Find PT and PR Solution: 4 x - = -because the sides are divided 7 12 proportionally when you draw a parallel line to another side. 7x = 48 x = 48/7 PT = 48/7 Cross products PR = 12 + 48/7 = 132/7 Congruency If two shapes are congruent, they are identical in both shape and size. Remember: Shapes can be congruent even if one of them has been rotated or reflected. The symbol means 'is congruent to'. Two triangles are congruent if one of the following conditions applies: 1. Three sides are the same The three sides of the first triangle are equal to the three sides of the second triangle (the SSS rule: Side Side Side). 2. Two sides and one angle are the same Two sides of the first triangle are equal to two sides of the second triangle, and the included angle is equal (the SAS rule: Side Angle Side). 3. Two angles and one side are the same Two angles in the first triangle are equal to two angles in the second triangle, and one (similarly located) side is equal (the AAS rule: Angle Angle Side). 4. Two sides in right-angled triangle are the same In a right-angled triangle, the hypotenuse and one other side in the first triangle are equal to the hypotenuse and corresponding side in the second triangle (the RHS rule: Right-angled, Hypotenuse, Side). Question For each of the following pairs of triangles, state whether they are congruent. If they are, give a reason for your answer (SSS, SAS, AAS or RHS). Pair 1 Pair 2 Pair 3 Answer 1. Yes. RHS 2. Yes. SSS 3. No. The side of length 7cm is not in the same position on both triangles. Therefore, it is not AAS. Similar areas and volumes We already know that if two shapes are similar their corresponding sides are in the same ratio, and their corresponding angles are equal. Look at the two cubes below: The cubes are similar, and the ratio of their lengths is a:b or Question What is the ratio of: 1. the area of their faces? 2. their volumes? Answer 1. Cube 'a' has a face area of a2 Cube 'b' has a face area of b2 The ratio of their areas is a2 :b2 or 2. Cube 'a' has a volume of a3 Cube 'b' has a volume of b3 The ratio of their volumes is a3 :b3 or For any pair of similar shapes, the following is true: Ratio of lengths = a:b or Ratio of areas = a2:b2 or Ratio of volumes = a3:b3 or Question These two shapes are similar. What is the length of x? Answer The ratio of the areas is 25:36 (a2:b2) The ratio of the lengths is a:b Therefore, we find the square roots of 25 and 36. Ratio of lengths = 5:6 5:6 = 2:x So (multiply both sides by 2) = x x = 2.4cm Now you try one. Question Two similar pyramids have volumes of 64cm3 and 343cm3. What is the ratio of their surface areas? Answer The answer is 16:49. Here is how to work it out. Try to fill in the blanks below: Ratio of volumes = 64:? To find the ratio of the lengths, we find the cube roots of 64 and ? Therefore, the ratio of the lengths is ?:7 To find the ratio of the areas, we square ? and 7 The ratio of the areas is 16:49. done by farhan grade 11d