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SIMILARITY Similar Polygons. Note that the two polygons to the left differ in size but are alike in shape.The two polygons are said to be similar. A formal definition of similar (~) polygons includes terms such as one-to-one correspondence, corresponding angles and corresponding sides. Two polygons are said to be similar if and only if there is a one-to-one correspondence between their vertices such that: 1. Corresponding angles are congruent (equal ()) 2. Lengths of corresponding sides are in proportion. The polygons shown are similar. When we say polygon CDEF is similar to polygon VWXY, we are asserting that the vertices have been paired as follows: C V D W E X F Y Y 1. Angle C angle V F E Angle D angle W V Angle E angle X Angle F angle Y C X W D 2. CD DE EF FC VW WX XY YV At this stage if we wish to show that two polygons are similar, we must establish that both conditions of the definition are met. The following figures show that meeting just one condition is not sufficient. 5 5 4 4 3 5 4 4 Polygon 1 Polygon 2 Polygon 3 Notice that although the corresponding angles of polygons 1 and 2 are congruent, the lengths of corresponding sides are not in proportion. Now look at polygons 2 and 3. In this case the lengths of the corresponding sides are in proportion but the corresponding angles are not congruent. Again the polygons are not similar. 1 R 6 W 3 A 4 12 6 D 8 S 2 T B 4 C The perimeter of a polygon is the sum of the lengths of its sides. Consider the perimeters of the similar quadrilaterals above. Note that the ratio if the perimeter of the quadrilateral RSTW to the perimeter of quadrilateral ABCD is 15:30 or 1:2. How does this ratio compare with the ratio of the lengths of each pair of corresponding side? 1. If two polygons are similar, the ratio of their perimeter equals to the ratio of the lengths of any pair of corresponding sides. Given: Polygon MORST is similar to polygon M’O’R’S’T’, the polygons have perimeter p and p’ respectively. T’ T M S M’ S’ O R O’ R’ p MO p' M ' O' Exercise 1 Find the unknown sides for each of the following: 1. C 4 O y x 5 B N P 2 3 z D M 6 C 2 2. T 10 y S C 4 x 6 B 3 A W z R D 7 3. I 8 y E 6 H D z 5 3 F J K G x 6 4. P x O D 8 3 C 4 M 5 5 N 2 A y B 5. The drawing shows a rectangular picture 16cm × 8cm surrounded by a border of width 4cm. are the two rectangles similar? 16 8 3 6. The diagonals of a trapezium ABCD intersect at O. AB is parallel to DC, AB = 3cm and DC = 6cm. If CO – 4cm and OB = 3cm, find AO and DO. 7. From the rectangle ABCD a square is cut off to leave rectangle BCEF. Rectangle BCEF is similar to ABCD. Find x and hence state the ratio of the sides of rectangle ABCD. ABCD is called the Golden Rectangle and is an important shape in architecture. A F B 1 D 1 E C Similar triangles. At this point in our discussion of similar polygons, the only way that we can rove two triangles similar is by showing that the triangles satisfy the definition of similar polygons. 1. If two angles in one triangle are congruent to two angles of another triangle, the triangles are similar (AA) Example: B D1 E 4 3 2 C F Given: Plane figure with angle 1 angle 2 Angle 3 angle 4 because they are vertically opposite angles. DE DF Because lengths of corresponding sides of similar triangles are in CE CB proportion. DEF ~ CED 2. If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides including these angles are proportional, the triangles are similar. C T A B R Given: ABC and RST with angle C angle T AC BC RT ST ABC ~ RST S 4 Example: A tree of height 4m casts a shadow of length 6.5cm. Find the height of a house casting a shadow 26m long. Tree House 6.5m 26m 4m x 6.5 4 26 x 26 × 4 = 6.5x 104 6.5 x x = 16 6.5 6.5 Therefore, the height of the house is 16m. Exercise 2 State whether the triangles are similar 1, 500 30 1000 0 2. 300 700 5 3. 300 a a b b 700 700 R 4. Complete the following question: a) Triangle RST ~ Triangle _______ b) Complete the extended proportion RS ST RT c) If RM = 3, MN = 4 and PS = 7, then ST = _______ d) If RM = 4, MN = 5 and ST = 8, then RS = _______ 1 M N 2 S T 5. Find the sides marked with letters in the following question. A Q x 16cm y 6cm B C P 3cm R 6cm 3. If a line is parallel to one side of a triangle and intersects the other two side, it divides them proportionally. Given: Triangle ABC; C Line YZ is parallel to line AB Angle 1 angle 2; Y 2 4 Z Angle 3 angle 4 AY BZ 1 3 YC ZC A B 6 Example: Given ABC with line YZ parallel to line AB Find the ratio of: a) CZ to ZB CZ CY 7 ZB YA 4 C 7 Y 4 b) BC to ZC BC AC 11 ZC YC 7 A Z B c) BC to BZ BC AC 11 BZ AY 4 Exercise 3 Find the sides marked with letters in questions 1 to 3; all lengths are given in centimeters. 1. 3. C X w 3 4 B 6 B y D 3 4 2 A C A x E 2 D 10 2. P m Q 2 R 5 3 T 3 S Summary: 1. If two convex polygons are similar: a) Corresponding angles are congruent b) Lengths of corresponding sides are in proportion c) The ratio of the perimeters equals the ratio of the lengths of any pair of corresponding sides. 7 2. Two triangles are similar if two angles of one triangle are congruent to tow angles of the other triangle. 3. In any triangle, a line that is parallel to one side and intersects the other two sides divides them proportionally. 4. It is important to note that some figures are a must to be similar. These figures include: Two equilateral triangles Two squares Two regular pentagons Two circles Areas of similar shapes The two rectangles are similar, the ratio of the corresponding sides being k. A B W area of ABCD = ab a area of WXYZ = ka × kb = k2ab ka 2 2 Area WXYZ = k ab = k C b D Area ABCD ab Z This illustrates an important general rule for all similar shapes: X kb Y If two figures are similar and the ratio of corresponding sides k, then the ratio of their areas is k2 Note: k is sometimes called the linear scale factor This result also applies for the surface area of similar three dimensional objects. Area scale factor = (linear scale factor)2 Example 1: XY is parallel to BC. AB = 3 AX 2 If the area of triangle AXY = 4cm2, find the area of triangle ABC. 3 2 The triangles ABC and AXY are similar. X Area scale factor = (linear scale factor) 2 4 = 22 B 2 x 3 x=9 A Y C 8 Example 2: Two similar triangles have areas of 18cm2 and 32cm2 respectively. If the base of the smaller triangle is 6cm, find the base of the larger triangle. Area scale factor = (linear scale factor) 18 = 62 32 x2 x = 8cm 18cm 32cm m 6cm x cm Example 3: A floor is covered by 600 tiles which are 10cm by 10cm. How many 20cm by 20cm tiles are needed to cover the same floor? Total area = 10 × 10 × 600 = 60000cm2 For 20cm by 20cm, 60000 20×20 Therefore, 150 tiles are needed to cover the same floor. Exercise 4: 1. Find the unknown area A. In each case the shapes are similar. a) 4cm2 A 3cm 6cm b) 2cm 3cm2 6cm A c) A 8cm 16cm 18cm2 16cm 3cm 9 d) A 27cm2 8cm 12cm 2. Find the lengths marked for each pair of similar shapes. a) 5cm2 4cm 20cm2 x b) 4cm2 9cm2 3cm z c) 12cm2 5cm 3cm2 a 3. Given Ad = 3cm, AB = 5cm and area of triangle ADE = 6cm2 Find: a) Area of triangle ABC b) Area of DECB A D E B 4. The triangles ABC and EBD are similar (AC and DE are not parallel) If AB = 8cm, BE = 4cm and the area of triangle DBE = 6cm2, find the Area of triangle ABC C A D B E C 10 5. A wall is covered by 160 tiles which are 15cm by 15cm. how many 10cm by 10cm tiles are needed to cover the same wall? 6. When potatoes are peeled do you lose more peel or less when big potatoes are used as opposed to small ones? Congruency Suppose we match the vertices of triangle ABC with those of triangle DEF in the following way A D B E C F This enables us to speak of correspondence between the triangles: Triangle ABC Triangle DEF C F A B D C In this correspondence, the first vertices named A and D, are corresponding vertices. So are the second and the third vertices named. Because A and D are corresponding vertices, angle A and angle D are called corresponding angles of the triangles. Other corresponding angles are angle B and angle E; angle C and angle F. Because vertices A and B correspond to vertices D and E, line AB and line DE are called corresponding sides. Other corresponding sides are line BC and line EF; line AC and line DF. When the above six statements are true for triangle ABC and triangle DEF, the triangles are said to be congruent ( ) triangles. There are several ways to find if triangles are congruent. These include: 1. If three sides of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (SSS) C F A B D E According to postulate: If line AB line DE, line BC line EF, and line AC line DF then ABC is DEF. 11 2. If two sides and the included angle of one triangle are congruent to the corresponding parts on another triangle, the triangles are congruent. (SAS) C F A B D E According to postulate: If line AB line DE, line AC line DF and angle A angle D then ABC is congruent to DEF Exercise 5 1. Fill in the blanks: a) Pair vertex A with vertex ______ b) Pair vertex C with vertex ______ c) Angle B and angle ______ are corresponding angles. d) Line CB and line _____ are corresponding sides. e) Which statement is correct, ABC KSV or ABC KVS? 15 10 C B 7 10 K S 7 A 15 V 2. Prove two triangles are congruent. a) Pair vertex D with vertex ______, and vertex C with vertex ______ b) Line DF and ______ are corresponding sides. c) Line EF and ______ are corresponding sides. d) Is the statement, DEF ZJO correct? e) Is the statement, OZJ FDE correct? F D O 1100 1100 E Z J 12 Some ways to find right triangles that are congruent are: 1. If two legs of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (LL) T R S Z X Y RST and XYZ; Angle S and angle Y are right angles. Line RS line XY; line ST line YZ RST XYZ. 2. If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (HL) According to postulate: If triangles RST and XYZ are right triangles with right angles S and Y, line RT XZ and line RS line XY then RST XYZ. Exercise 6 1. In the figure it is given that line XC line AE, line AX line XD and line BX line XE. Name a right triangle that has: a) Hypotenuse XD X b) Hypotenuse XE c) Line XD as one of its legs d) Line BX as one of its legs e) Line AS as its hypotenuse. f) Name every right triangle that has XC as one of its legs. A B C D E 13 2. Name each right triangle shown in the figure: a) Given: Angle ACD is a right angle; line CD is perpendicular to line AB C A D B b) Given: Angle ABC, angle BCD, angle CDA and angle DAB are right angles. D C E A B More ways to find triangles congruent are: 1. If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (ASA) C A F B D E According to postulate: If angle A angle D, line AB DE, and angle B angle E; then ABC DEF. 14 2. If two angles and a not- included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (AAS) C F D A E B According to theorem: Triangles ABC and DEF; Angle A angle D; angle C angle F; line AB line DE. ABC DEF. Exercise 7 1. Name the side that is included between the angles named: a) A, B b) B, C A c) A, C C B 2. Name the two sides that are not included between the angles named: a) D, E D b) E, F c) D, F F E More ways to find right triangles congruent are: 1. If the hypotenuse and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (HA) 15 According to theorem: Triangles ABC and DEF; angle B and angle E are right angles; line AC line DF; angle A angle D ABC is congruent to DEF C A B F D E 2. If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. Case 1 Triangles RST and XYZ; angle S and angle Y are right angles; line RS line XY; angle R angle X T Z RST XYZ. R S X Y Z Case 2 Triangles RST and XYZ; angle S and angle Y are right angles; line RS line XY; angle T angle Z. RST XYZ. T X R Y S 16 Exercise 8 State whether the HA theorem, the LA theorem, or some other right triangle method can be used to prove that the triangles are congruent. 1. 2. 3. 4. 17 5. Summary 1. Ways to prove two triangles congruent: All triangles Right triangles SSS ASA LL HA SAS AAS HL LA 2. A common way to prove that two angles or two segments are congruent is to show that they are corresponding parts of congruent triangles. 3. If two sides of a triangle are congruent, the angles opposite those sides are congruent. 4. If two angles of a triangle ate congruent, the sides opposite those angles are congruent. Similar 3D shapes Volume of similar objects: When solid objects are similar, one is an accurate enlargement of the other. If two objects are similar and the ratio of corresponding sides is k, then the ratio of their volumes is k3. A line has one dimension, and the scale factor is used once. An area has two dimensions, and the scale factor is used twice. A surface area of 3 dimensions figures also uses the scale factor twice. A volume has three dimensions, and the scale factor is used three times. Volume scale factor = (linear scale factor)3 18 Example: 1. 6cm 3cm 30cm3 Two similar cylinders have heights of 3cm and 6cm respectively. If the volume of the smaller cylinder is 30cm3 , find the volume of the larger cylinder. If the linear scale factor = k, then ratio of heights (k) = 6 =2 3 ratio of volumes (k3) = 23 =8 And volume of larger cylinder = 8 × 30 = 240cm3 2. Two similar spheres made of the same material have weights of 32kg and 108 kg respectively. If the radius of the larger sphere is 9cm, find the radius of the smaller sphere. We may take the ratio of weights to be the same as the ratio of volumes. 32 108 8 = 27 Ratio of volumes (k3) = Ratio of corresponding lengths (k) = 3 8 27 2 3 2 Radius of smaller sphere = ×9 3 = 6cm = 19 Exercise 9 In this exercise, the objects are similar and a umber written inside a figure represents the volume of the object in cm3. Numbers on the outside give linear dimensions in cm. Find the unknown volume, V for questions 1- 4 1. 60 V 5 10 2. 4.5 V Radius = 1.2 Radius = 12cm 3. V 54 8 12 4. 88 V 6.1 3.1 20 In questions 5 and 6, find the lengths marked by a letter. 5. 54 16 6 m 6. 7 10 y 270 7. Two spherical balls are made with the same material. Their masses are 30kg and 18kg respectively. If the radius of the smaller sphere is 3cm, find that of the larger one. 8. Two similar cylindrical tins have base radii of 6cm and 8cm respectively. If the capacity of the larger tine is 252cm3, find the capacity of the small tin. 9. Two similar cones have surface areas in the ratio 4:9. Find the ratio of: a) their lengths, b) their volumes. 10. A container has a surface area of 5000cm2 and the capacity is 12.8 litres. Find the surface area of a similar container which has a capacity of 5.4 litres. 21 Answers: Exercise 1 1 2 1. x = 6 , y = 3 , z = 4 3 3 4 1 4. x = 4 , y = 8 5 5 7. 0.618 ; 1.618 :1 2. x = 5, y = 6 2 2 , z = 11 3 3 1 1 2 3. x = 5 , y = 6 , z = 2 3 5 2 5. No 6. 2cm, 5cm 1. Yes 2. No 3. No 4. (a) RMN (b) RM, MN, RN (c) 13 Exercise 2 (d) 6 2 5 1 3 5. x = 12cm, y = 8cm Exercise 3 1. y = 6 1 4 2. m = 5 , z = 4 3 5 3. x = 4, w = 1 1 2 Exercise 4 1. (a) 16cm2 (d) 12cm2 1 cm 2 4. 24cm 2 (c) 2 (b) 27cm2 2. (a) 8cm 3. (a) 16 2 cm 2 3 5. 360 (c) 128cm2 1 (b) 4 cm 2 2 (b) 10 cm2 3 6. Less (for the same weight) Exercise 5 1. (a) K (d) line SV (b) line OZ (e) Yes Exercise 6 1. (a) XCD (d) BXE (b) S (e) ABC KVS (c) line OJ (c) V 2. (a) Z, J (d) Yes (b) XCE (e) AXD (c) AXD (f) ACX, XCE, XCD, XCB 2. (a) ACB, CDB, ADC (b) ABC, BCD, CDA, AEB, DAB, DEC, BEC, AED 22 Exercise 7 1. (a) line AB 2. (a) line DF, EF (b) line BC (b) line DE, DF (c) line AC (c) line DE, EF 2. AA 5. HA 3. LA Exercise 8 1. LA 4. HA Ravina, you have done well overall..but could have given more examples and some real life problems… GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCY Writing Drawin skill g skill 3 2 creativit picture y flow s neatnes labelin s g 3 2 Formatti ng skill 3 page paragrap hs math equations 3 Polygo n 2 alwayssimilar polygo ns appln 1 Triangle 2 theore ms applns 3D shapes 3 perimet er vol, SA 2 3 Exampl Questions es 3 2 3D ex+ans shapes congruen cy variety 1 1 Total 20 16 23