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SIMILARITY AND CONGRUENCY. Two figures (e.g. polygons) are similar when they both have the same shape, and one object is congruent (equal) to the uniform enlargement of the other. The corresponding (i.e. equivalent) sides of similar polygons are in proportion, and corresponding angles of similar polygons are equal in measure. In short, this means that the figures have the same shape but not the same size. Below, the shapes shown in the same colour are similar: The rectangles below are similar to each other: Many times, you will be asked to find the angles and sides of figures. Similar polygons can help you out: 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 multiplication to solve. 4 x 6 9 4 7 6 y x=6 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 triangles have a couple of special rules dealing with similarity. They are: Similarity rules for triangles: 1. Angle-Angle Similarity (AA): If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. 2. Side-Side-Side Similarity (SSS): If all pairs of corresponding sides of two triangles are proportional, then the triangles are similar. 3. Side-Angle-Side Similarity (SAS): 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. 1) 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) 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. NOTE: Some figures will always remain similar. These are squares, circles and equilateral triangles. The reason for them to remain similar is that the ratio of the sides is always 1 (e.g. in squares the ratio of the sides is always 1:1:1:1) EXERCISE 1: The following figures are similar. Find the sides marked with letters. (in cm) 1) 2 a 5 e 4 6 2) 9 x 8 12 15 y 3) x 16 6 y 6 3 AREAS OF SIMILAR SHAPES The two rectangles below are similar, and the ratio of the corresponding sides is ‘k’. (NOTE: ‘k’ is sometimes called the scale factor) W A X B A ka D b C Z Y kb Area of ABCD= ab Area of WXYZ=k2ab AreaWXYZ k 2 ab = 2 AreaABCD ab k This shows an important rule for all similar figures: if two figures are similar and their ratio is ‘k’, then the ratio of their areas will be ‘k2’ A XY is parallel to BC. 2 3 X B Y C If the area of triangle AXY=4cm2, find the area of triangle ABC. Solution: Triangle ABC and AXY are similar. The ratio of corresponding sides (k) = Ratio of areas (k2) = 3 2 9 4 Area of triangle ABC= = 9 × (area of 4 AXY) 9 × 4 =9 cm2 4 CONGRUENT FIGURES Two-dimensional figures are congruent only if one of the figures fits exactly on the other. This means that they must be the same shape and size. The following shapes are congruent: CONDITIONS FOR CONGRUENT FIGURES. Two triangles are congruent if you can show one of the following conditions: There are 4 conditions: SSS (side,side,side): if 3 sides of one triangle is equal to 3 sides of another triangle, then the 2 triangles are congruent. SAS (side,angle, side): if 2 sides of one triangle is equal to 2 sides of another triangle and the angle between them is equal on both triangles, then the 2 triangles are congruent. ASA (angle,side,angle): if 2 angles on one triangle are equal to 2 corresponding angles on the other triangle and the length of the side between the pair of angles is equal on both sides, then the triangles are congruent. RHS (right angle, hypotenuse, side): if two right-angled triangles have their *longest side equal in length and another side of the first triangle is equal to a side of the other triangle, then the two triangles are congruent. NOTE: the longest side of a right-angled triangle is always opposite the right angle. This side is known as the hypotenuse of the triangle. EXERCISE 2: In each of the following cases, state whether the two triangles are congruent and, if they are, give a reason. (the diagrams are not to scale.) a) 85º 45º 3 cm 45º 3 cm 85º b) 50º 3 cm 3 cm 50º 5 cm 5 cm c) 80º 3 cm 3 cm 60º 40º 4 cm 4 cm d) 3 cm 7 cm 7 cm 3 cm 3-D SHAPES. 3-D shapes are similar when one shape is an exact enlargement of the other. A line has one dimension, and so the scale factor (k) is used once An area has two dimensions, and so the scale factor is used twice (k2) A volume has three dimensions, and so the scale factor is used three times (k3) Two similar cylinders have heights of 3cm and 6cm respectively. If the volume of the smaller cylinder is 30 cm3, find the volume of the larger cylinder. 3 cm 30cm2 6 cm Solution: Ratio of heights (k) = 6 =2 3 Ratio of volume (k3) = 23 =8 Volume of large cylinder = 8×30 = 240 cm3 Two similar spheres made of the same material have weights 32kg and 108kg respectively. If the radius of the larger sphere is 9cm, find the radius of the smaller sphere. 9cm 32kg (108kg) Solution: We may take the ratio of weights to be the same as the ratio of volumes. 32 108 8 = 27 Ratio of volumes (k3) = 8 27 2 = 3 Ratio of corresponding lengths (k) = 2 ×9 3 =6 Radius of smaller sphere = 3 EXERCISE 3: In this exercise, all the objects are similar and the number written inside the figure represents the volume of the objects (in cm3) The numbers on the outside give the dimensions (in cm) Find the volume: 1) 60 V 5 cm 10 cm 2) 5 cm 20 15 cm V 3) 4.5 V Radius = 1.2 cm Radius = 12 cm 4) V 8 cm 54 12 cm ANSWERS: Exercise 1: 1) a= 2 1 , e= 3 cm 2 2) x= 6cm, y= 10 cm 3) x= 12 cm, y= 8 cm Exercise 2: ‘a’, ‘c’ and‘d’ are correct. ‘a’ is correct because it follows the ‘AA’ similarity. (Angle-Angle similarity). ‘c’ is correct because it follows the ‘SAS’ similarity (SideAngle-Side similarity). ‘d’ is correct because follows the ‘RHS’ similarity (Right angle, Hypotenuse, Side similarity) Exercise 3: 1) 480 cm3 2) 540 cm3 3) 4500 cm3 4) 16 cm3 Shaista, read the instructions properly.. you haven’t discussed about surface area, perimeter and similarity of different polygons. GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCY Writing Drawin skill g skill 3 2 creativit picture y flow s neatnes labelin s g 2 2 Formatti Polygo ng skill n 3 2 page alwaysparagrap similar hs math polygo equations ns appln 3 1 Triangle 2 theore ms applns 3D shapes 3 perimet er vol, SA 2 1 Exampl Questions es 3 2 3D ex+ans shapes congruen cy variety 2 1 Total 20 14