Similar Triangles. reason deductively and prove riders based on the theorems. analyse and solve real life problems based on the theorems.

Transcription

1 10 Similar Triangles Similar polygons and similar triangles. asic proportionality theorem. onverse and corollaries of T. and SSS criteria of similar triangles. roof of criteria of similar triangles. Theorem on areas of similar triangles. This unit facilitates you in, explaining the meaning of similar polygons and similar triangles. di fferenti ating betwe en similar triangles and congruent triangles. stating and proving basic proportionality theorem (T). stating and proving converse and corollaries of T. stating and proving criteria for similar triangles. stating the SSS and SS criteria for similar triangles. stating and proving theorem related to right angled triangle whe re perpendicular is drawn from the right angled vertex to the hypotenuse. stating and proving the theorem on areas of similar triangles. Thales of Miletus ( , Greece) Thales was the first known philosopher and mathematician. He is credited with the first use of deductive re asoning in geometry. He discovered many propositions in geometry. He is believed to have found the heights of the pyramids in gypt, using shadows and the principle of similar triangles reason deductively and prove riders based on the theorems. analyse and solve real life problems based on the theorems. The universe cannot be read until we have learnt the language in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which it is humanly impossible to comprehend a single word. - Galileo Galilei

2 30 UNIT-10 In your previous classes, you have learnt about triangles and their properties. These properties are used to solve many day-to-day life problems. Some of them are applied in studying other subjects like hysics, hemistry etc. elow are given some examples where triangles and their properties are involved. 1. s( s a)( s b)( s c) Where s = a + b + c. c b This is the formula to find the area of a triangle. o you know how this formula is derived?. = 14cm We want to divide in the ratio : 5. How can this be done practically? feet 3. There is an arch above the door. The width of the door is 3 feet and the height of the arch is feet. How to find the radius of the arch? u v f This is called the mirror formula, which you have learnt in physics. Have you ever thought how this formula is derived? N or all these queries and many more, you will find solutions in this unit. onsider the following example: Two groups of students did the following project. One group of students measured the length of the shadow cast by a tree. t the same time and next to the tree another group of students measured the length of the shadow cast by a vertical pole of three metres height which was next to the tree. With this information, they wanted to find the height of the tree. Is this possible?

3 Similar Triangles 31 Yes, it is possible to find the height of the tree using an important concept from Geometry called "Similar Triangles". In order to understand about similar triangles, let us first learn an important i dea abou t shape and size o f geometrical figures. The geometrical figures on a plane with reference to their shapes and sizes can have three possibilities : Know this! Historians tell us that, Thales Greek Mathematician, about 600. found the height of yramid in gypt on the basis of the length of its shadow. ossibility - 1 ossibility - ossibility - 3 The figures having neither The figures having the The figures having the same the same shape nor the same shape and shape but not the size. same size. same size. square and a rhombus Two triangles having Two equiangular triangles same measurements. having different measurements 40º 50º These are called congruent figures 50º 100º 30º These are called similar figures. Like congruence, similarity relation also plays an important role in geometry. ven in our daily life, concept of similarity is used very widely. or example, the maps used in Geography are based on the concept of similarity. Models are made before constructing big buildings which are similar. We have already studied about congruent triangles. 'Two triangles having the same shape and same size are congruent triangles. Now let us study more about the figures which have the same shape and not necessarily the same size, which are called similar figures. Similar figures Observe the given photographs. You can at once say that they are the photographs of the same building (Vidhana Soudha) but are in different sizes. re these photographs similar? iscuss. Now, let us consider some similar geometrical figures Two circles are always similar S S Two squares are always similar Two equiangular triangles are always similar

4 3 UNIT-10 rom the above examples, we can conclude that "two figures are similar if and only if they have same shape, but not necessarily the same size". We use the symbol (read as similar to) to indicate the similarities of figures. or example: efer to the third set in the above figures. is similar to Know this! Similar, means having same shape. If we have two figures and out of these two, one can be obtained either by diminishing (shrinking) or by enlarging (stretching) the other without any change in their shape, then the two figures are similar. This has happened to the photographs of Vidhana Soudha and hence they are similar. In all the examples discussed so far, we have assumed the figures to be similar by appearance. ut, how to express similarity of figures mathematically. We need some definition of similarity and based on this definition some rules or conditions are evolved to decide whether the two given figures are similar or not. Now let us study about the similarity of polygons and its definition. Similar olygons onsider the pairs of similar geometrical figures given below. In each case observe the corresponding angles and the ratios of corresponding sides. airs of similar Geometrical figures The other symbol for similarity is Observations In and S 1.,,, S and S. S S S T S In and ST 1.,,, S, T S ST T ST

5 Similar Triangles 33 rom the above examples, we can infer that, "Two polygons of same number of sides are similar, if condition 1 : all the corresponding angles are equal. and condition : all the corresponding sides are in the same ratio or in a proportion. iscuss in groups Which are the geometrical figures having same number of sides, that : (a) may or may not be similar? (b) can never be similar? ctivity : S S O ut out some geometrical shapes like triangles or quadrilaterals from a piece of card board. Hold these cutouts one by one, between a point source of light and a wall. Look at the shadow cast by each cutout on the wall. The shadows have the same shape as the original cutouts, but are larger in size. The shadows are said to be similar to the original cutouts. In the figure S is a quadrilateral and the quadrilateral I I I S I is its shadow. The quadrilateral I I I S I is an enlargement (or magnification) of the quadrilateral S. Note that I lies on the ray O, I lies on the ray O. I lies on the ray O and S I lies on the ray OS. lso note that Vertex I corresponds to, it is written as I Vertex I corresponds to, it is written as I Vertex I corresponds to, it is written as I Vertex S I corresponds to S, it is written as S I S We can also observe that, 1. = I = I = I and S = S I. I I I I I I S I I S S S Note: This ratio of the corresponding sides is referred to as the scale factor for the polygon. Know this! Hills Law: nimals which are geometrically similar, jump to the same height. rystalline Silicon is isomorphous with diamond i.e., they have similar structures.

6 34 UNIT-10 Similarity of Triangles ecall that any two congruent triangles have six pairs of corresponding elements equal, ie., three pairs of corresponding angles and three pairs of corresponding sides. ut while determining the congruency of any two triangles we have obtained some criteria involving only three pairs of the corresponding elements of the two triangles. They are SS, SSS, S and HS riteria. We have also found that, when two triangles are equiangular ( criteria) they may or may not be congruent triangles. Observe the following examples. xample 1: X X, Y, Z = XY, = YZ, = XZ Y Z xample : = X, = Y, = Z; XY, YZ, XZ In example 1 : ll the 3 pairs of corresponding angles are equal and all the 3 pairs of corresponding sides are equal. is congruent to XYZ i.e. XYZ In example : ll the 3 pairs of corresponding angles are equal and all the 3 pairs of corresponding sides are not equal. is not congruent to XYZ. X Y Z If two triangles are equiangular and they are not congruent, their corresponding sides may or may not be equal. Then, how are their corresponding sides related to each other? Observe the pairs of equiangular triangles given below. Write the measures of their corresponding sides in the table. ind their ratios as shown..5 cm 4 cm 3 cm 5 cm 6 cm 8 cm 9 cm 6 cm 1 cm cm 4 cm 3 cm

7 Similar Triangles 35 quiangular triangles atio of corresponding sides 1. and.5 5 = =. and = = = In each of the above examples, what can you conclude about the ratios of the corresponding sides? We observe that the ratios between corresponding sides are equal. If the ratios are equal, then the corresponding sides are said to be in proportion. We can conclude that, If two triangles are equiangular, then their corresponding sides are in proportion. This can be symbolically represented as follows. In and 1. = = =. Triangles are the only type of polygons, where the two conditions mentioned in the definition of similar figures need not be fulfilled. If any one of these conditions is fulfilled, then the triangles are similar. Two triangles are said to be similar, if their corresponding angles are equal. or their corresponding sides are proportional. Now let us discuss about the corresponding angles and sides of similar triangles. orresponding angles: In two similar triangles, angles which are equal are called corresponding angles. If, ~ = as they are opposite to corresponding, sides and respectively. Similarly = and = orresponding sides: In similar triangles, sides opposite to equal angles are known as corresponding sides and they are in a proportion. In the figure =, =, =

8 36 UNIT-10 and are corresponding sides as they are opposite to the equal angles and respectively. Similarly, and are corre sponding sides as they are opposite to and respectively and and are corre spon ding sides as they are opposite to and respectively. Thus, which are in a proportion. ongruency and similarities of triangles are corresponding ratios between sides of similar triangles ongruency is a particular case of similarity. In both the cases, three angles of one triangle are equal to the three corresponding angles of the other triangle. ut in congruent triangles, the corresponding sides are equal, while in similar triangles, the corresponding sides are proportional. ongruent triangles Similar triangles =, =, = = ; = ; = ~ =, =, = ; ; 1 Same shape and same size 1or < 1 Same shape but not same size. We can also conclude that, 'ongruent triangles are always similar.' but 'Similar triangles are not necessarily congruent'. undamental roperties of similar triangles

9 Similar Triangles 37 roperty 1 (This is with regard to angles) "If two triangles are similar, then the measure of three angles of one triangle are equal to measure of corresponding three angles of another triangle". If then, = = = roperty (This is with regard to sides) "If two triangles are similar, the three sides of one triangle are proportional to the corresponding three sides of another triangle". If then onverse of roperty 1 and "If the measure of three angles of one triangle are e qual to measure of corresponding three angles of another triangle then, the two triangles are similar". If, in and = = = "If three sides of one triangle, are proportional to the corresponding three sides of another triangle then the two triangles are similar". If, in and then, then, We can combine property 1 and as follows "If two triangles are similar, then and * they are equiangular * the corresponding sides are in proportion. "If two triangles are similar, then and * the corresponding sides are in proportion * they are equiangular. ILLUSTTIV XMLS Numerical problems based on similarity of triangles. xample 1 : In the adjoining figure TS. Identify the corresponding vertices, corresponding sides and their ratios. S Sol. Given : TS T

10 38 UNIT-10 The corresponding Vertices The corresponding Sides T S S T S T The ratios are ST S T xample : rom the following data, state if or not. (a) = 65 0, = 8 0, = 33 0, = 65 0 (b) =5cm, =7cm, =70 0, =70 0, =15 cm, =8 cm. Solution : (a) In, = 65 0, = 8 0. = ( ) = = In, = 33 0, = 65 0 = ( ) = = 8 0. =, = and =. (b) In and, = = is not similar to. xample 3 : In, XY. If XY = 3cm, Y = cm, and = 6cm, find the length of. Solution: Given :- XY = 3 cm, Y = cm, = 6cm and XY. In XY and, XY ( orresponding angles) YX is common. ( orresponding angles) X cm Y 3cm 6cm XY

11 Similar Triangles 39 orresponding sides are proportional. X XY Y XY Y XY Y cm. xample 4 : girl of height 90 cm is walking away from the base of a lamp - post at a speed of 1. m/s. If the lamp is 3.6m above the ground, find the length of her shadow after 4 seconds. Sol. Let, denote the lamp - post. = 3.6m denote the height of the girl. = 90 cm = 0.9m ( 90 cm = 0.9 m) The girl is walking for 4 seconds away from the lamp - post, at a speed of 1.m/s the distance covered is 4 1. = 4.8m = 4.8m denote the shadow cast by the girl. Let = x metres. In and, = = 90 0 ( lamp-post and the girl are standing vertical to the ground) = ( ommon angle) ( ngle sum property of triangle) 4.8+ x 3.6 ; = ; x = 4x ; 3x = 4.8 ; x = 1.6 x 0.9 The shadow of the girl after walking for 4 seconds is 1.6m long. XIS In the given pairs of similar triangles, write the corresponding vertices, corresponding sides and their ratios. L Z Y (a) K (b) M X ( c)

12 40 UNIT-10. Study the following figures and find out in each case whether the triangles are similar. Give reason G 60 H 55 a 40 a (ii) a X 6 cm M cm6 cm 4 cm a U a b c V c N cm 8 cm O Y 3 b Z (iii) (iv) 3. ind the unknown values in each of the following figures. ll lengths are given in centimetres. (Measures are not to scale) x y G N 3 5 M 6 4 S y 6 x H J a 1 G b 4 K L b K 6 (i i ) (ii) (iii) 4. In the given figure, l 1 l show that O O. l 1 0 l 5. rom the follwing data, state whether is similar to or not. (a) = 70 0, = 80 0, = 70 0, = 30 0 (b) =50 0, =50 0, =5cm, =6cm, =.5cm, =3cm. (c) =8cm, =9cm, =15 cm, =4cm, =3cm, =5cm. (d) =16cm, =4cm, =4cm, =6cm, = 6 0, = 6 (e) = 5 0, = 5 0, = 5cm, = 30 cm, = 5cm, = 7cm.

13 Similar Triangles In, (a) If = 6cm and = 18cm, find :. (b) If = 0cm, =4cm and = 3cm, find. (c) If = 3cm, = 15cm, =4cm, find. (d) If = 3.5cm, = 7cm, = 8cm, find. (e) If = 1cm, = 3cm, = 16cm, find. 7. In the given figure,, = 7cm, = 5cm, = 4cm. If = 1cm, find and. 8. In XYZ, is any point on XY and XZ If X = 4cm, XY = 16cm and XZ = 4cm, find X. 1cm 5cm 7cm X 4cm Y Z 9. Select the set of numbers in the following, which can form similar triangles. (i) 3,4,6 (ii) 9, 1, 18 (iii) 8, 6, 1 (iv) 8, 4, 9 (v), 4½, vertical pole of 10m casts a shadow of 8m at certain time of the day. What will be the length of the shadow cast by the tower standing next to the pole, if its height is 110 m? 11. ladder resting against a vertical wall has its foot on the ground at a distance of 6ft. from the wall. man on the ground climbs two - thirds of the ladder. What will be his distance from the wall now? undamental geometrical result on proportionality: ctivity: Study the following triangles carefully. In, is drawn parallel to. Observe the measures of the line segments,, and. Now consider the ratios of the two sides which are divded by. What can you conclude about the ratios? iscuss in groups. In, cm 6 cm 1.5 cm 4.5 cm In, cm ig

14 4 UNIT-10 In, 4 cm 5 cm cm 5 cm cm ig. 1 In, 6 cm 4 cm cm 5 cm ig. 3 cm cm 4 cm 6 cm 5 cm ig. 4 cm 4 In, is not parallel to the side In the first three triangles, each of the ratios are equal but in the fourth triangle, the ratios are not equal. So in a triangles, what is the important condition / criteria that makes the ratios equal? The only important condition is that in, if is parallel to, then the ratios and will be equal. The above result can be generalized and it is called asic roportionality Theorem (..T) or Thales Theorem. It can be stated as, "If a straight line is drawn parallel to one side of a triangle, then it divides the other two sides proportionally". Now let us logically prove the Thales theorem.

15 Similar Triangles 43 Thales Theorem : [asic proportionality theorem] If a straight line is drawn parallel to a side of a triangle, then it divides the other two sides proportionally". ata: In L N To prove: onstruction: 1. Join, and,. raw L and N roof: Statement eason rea of = rea of 1 L 1 L 1 = b h rea of = rea of 1 N 1 N 1 = b h and xiom -1 While proving Thales Theorem, the figure can be drawn in two more ways. This is because, the straight line drawn parallel to a side of a triangle can be outside the triangle also. (above the triangle or below the triangle). Study the following figures and with the help of the same constructions try to prove them individually. ig. 1 ig. In figure 1, the line is above the vertex, where as in figure the line is below the base line. Note that '' is the point where the parallel line cuts, or produced or produced. Similarly the point is the point where the parallell line cuts or prduced or produced.

16 44 UNIT-10 onverse of Thales Theorem "If a straight line divides two sides of a triangle proportionally, then the straight line is parallel to the third side". ata: In, To prove: onstruction: raw roof: Statement eason If is not parallel to, then let be parallel to [ Thales Theorem] ut [ ata] = oint coincides point xiom1 Hence, Now, let us study about some corollaries of Thales theorem. orollary 1: In ; dd 1 to both the sides, [..T Theorem] 1 1 ; This is called componendo orollary : In, [ T theorem] Taking reciprocals, we get dd 1 to both the sides. 1 1 This is also called componendo

17 Similar Triangles 45 orollary 3: In, and is a parallelogram [ definition of parallelogram] = In a parallelogram opposite sides are equal. In, [ corollary ] In, [ corollary 1] [ = ] orollary 3 can be stated as follows :- If a straight line is drawn parallel to a side of a triangle then the sides of intercepted triangle will be proportional to the sides of the given triangle. Numerical problems based on Thales theorem ILLUSTTIV XMLS Note : onverse for all the three corollaries exists and are true. 1. In,, = 5.7 cm, = 9.5cm, = 6cm. ind. Sol. Given: In, = 3.6cm 9.5. In the adjoining figure XY II. ( T) 9.5 cm. 5.7 cm.? 6 cm. X = p-3, X =p- and Y = 1. ind 'p'. Y 4 Sol. Given : In, XY X Y X Y ( Thales theorem) 4 p 3 1 p 4 1 Y 4 (p-3) = 1(p-), 4p-1 = p-, p = 10 p = 5 p 3 X p

18 46 UNIT In, and =. 3 Sol. Given: In,. If = 3.7cm, find. 3.7 cm ( Thales theorem)? = 5.5 cm. 4. In, S and T are two points on and respectively such that S = 4cm, S = 3cm, T = 6cm, T = 4.5cm and ST = ind. Sol. In, S = 4 S 3 T 6 4 T cm 6 cm S S T T ST = = 40 0 ST ( onverse of Thales theorem) 5. t a certain time of the day, a man 6 feet tall, casts his shadow 8feet long. ind the length of the shadow cast by a building 45 feet high, at the same time. Solution : Let the length of the shadow of the building = x. Since,. height of the man length of shadow cast by the man i.e., = height of the building length of shadow cast by the building 3 cm S? 40 0 T 4.5 cm 6 ft 8 ft 8 45 x 60 ft. 45 ft x 6 Length of shadow cast by the building is 60 ft. 45ft 6ft XIS 10. 8ft x 1. Study the adjoining figure. Write the ratios in relation to basic proportionality theorem and its corollories, in terms of a, b, c and d. d Y c a X b

19 Similar Triangles 47. In the adjoining figure,, = 7cm, = 5cm and = 18cm. ind and. 3. In, S is a point on such that ST and S 3 =. S 5 If = 5.6cm, find T. S T 4. In, and are points on the sides and respectively such that. (i) If = 6cm, = 9cm, and = 8cm, find. (ii) If = 8cm, = 1cm, = 1cm, find. (iii) If = 4x-3, = 3x-1, = 8x 7, and = 5x 3 find the value x. 5. In the figure, = 3cm, = 4.5 cm and = 5 cm. ind the length of. 6. In, and are points on the sides and respectively. or each of the following cases, verify. (i) = 3.9cm, = 3cm, = 3.6cm, =.4 cm. (ii) = 4cm, = 4.5cm, = 8cm, = 9cm. (iii) = 1.8cm, =.56cm, = 0.18cm, = 0.36cm. 7. Which of the following sets of data make G? (i) = 14cm, = 6cm, = 7cm, G = 3cm. (ii) = 1cm, = 3cm, = 8cm, G = 6cm. G (iii) = 6cm, = 5cm, G = 9cm, G = 8cm. 8. In the adjoining figure, and. O If O = 1cm, = 9cm, O = 8cm and = 4.5cm, find O.

20 48 UNIT In the figure, K and HK. If = 6cm, H = 4cm, H = 5cm and K = 18cm, find K and. H K 10. t a certain time of the day a tree casts its shadow 1.5 feet long. If the height of the tree is 5 feet, find the height of another tree that casts its shadow 0 feet long at the same time. ractical application of Thales Theorem : 1. Let us consider the problem discussed in age 30. To divide a line in a given ratio. ivide the line segment = 14 cm in the ratio : 5 N M cm 5 cm 4 cm 14 cm 10 cm Steps : 1. raw = 14 cm. raw to make an angle of any measure with [or convenience let < 60 0 ] 3. Mark points M and N on such that M = cm, MN = 5cm. 4. Join N and 5. raw M N using set squares 6. Measure and. = cm = cm. The above result can be proved in the following few steps using Thale's theorem. ata: In N, M, and M N MN 5 To prove: roof: 5 In N, M N M = MN 5 [ Thales Theorem] Now let us solve some riders based on Thales theorem.

21 Similar Triangles 49 ILLUSTTIV XMLS 1. In M = 3 N and = 3 rove that MN is a trapezium roof : (i) M 3N 3 M N [ ata] M N MN [ onverse of corollary of..t] (ii) Since and intersects at '', cannot be parallel to MN is a trapezium [ If only one pair of opposite sides of a quadrilateral is parallel then it is a trapezium]. rove that "In a trapezium, the line joining the mid points of non-parallel sides is (i) parallel to the parallel sides and (ii) Half of the sum of the parallel sides" ata : In the trapezium (i) (ii) X = X Z (iii) Y = Y To rove: (i) XY II X Y O XY II (ii) 1 XY ( ) onstrution :1. xtend and to meet at Z.. Join and. Let it cut XY at roof: Step 1: In Z, Z Z [ ata] [ T ] Z X Z Y [ X & Y are mid points of and ] Z Z X Y XY ---(i) [ onverse of..t]

22 50 UNIT-10 Step : In, 1. X = X [ ata]. X [ roved] = [ onverse of mid point theorem] X = ½ [ M. Theorem] ly In, Y = ½ y adding, we get X + Y = ½ + ½ XY = ½ ( + ) --- (ii) This problem can also be solved by doing alternate construction. Z i.e. instead of joining, draw II. Let it cut XY at. Now, discuss in groups and write the proof. X Y 3. In LMK, rove that ac x = b + c L ata:- In LMN, LMN = NK = 46 To prove:- LM = a, N = x, MN = b, NK = c ac x = b + c Statement eason roof:- In LMK, LM N If corresponding angles N NK LM MK x c = a b + c are equal then lines are parallel. orollary to Thales. a 46º x 46º M b N c K ac x =. b + c 4. hombus is inscribed in such that is one of its angle., and lie on, and respectively. If = 1 cm and = 6cm find the sides of rhombus. Statement eason Solution:- In, In a rhombus opposite sides are parallel. orollary to Thales.

23 Similar Triangles x x 6 1x = 7 6x 18x = 7 x = 4 cm The sides of the rhombus is 4cm each. x x x iders based on Thales theorem. XIS In,. calculate 'x' or for what value of 'x', will be parallel to x x 4. X is any point inside. X, X and X are joined. '' is any point on X. If, G. rove that G. (Hint:..T & xiom 1). x X G 3. State 'Mid point Theorem'. rove the theorem using 'onverse of Thale's Theorem. 4. In, =, and are the points on and such that, prove that. 5. In, & are the points on and such that =. rove that 6. is a quadrilateral in which W, X, Y and Z are the points of trisection of sides,, and respectively. rove that WXYZ is a parallelogram. (Hint: Join, ) Z 1 Y 1 1 X 7. is a trapezium in which iagonals 1 W intersect at 'O'. Show that O O O (Use T to prove it) O (Hint: raw through 'O' and apply O T to and

24 5 UNIT In, and rove that 9. In the figure, and rove that 10. In, and rove that = riteria for similarity of triangles ecall that we have certain criteria for determining the congruency of triangles. In the same way what is the criteria for determining the similarity of triangles? Is it necessary to check all the six pairs of corresponding elements of two triangles? let us study the criteria for similarity of two triangles. Let us discover these criteria through activities. ngle-ngle () similarity criterion for two triangles You are familiar with construction of triangles for given measurments. onsider the following activity. Two line segments and of lengths 4 cm and 5 cm respectively are drawn. t and angles = 60 and = 40 are constructed. In the same way, = 60 and S = 40 are constructed. riteria means a standard which is established so that judgement or decision, especially a scientific one can be made. S cm Let the rays and intersect at and rays and S intersect at. Now, we have the triangles and. Measure and. What do you observe? In and, you can see that =, = and =. That is, the corresponding angles of the two triangles are equal. What can you say about their corresponding sides? Note that, the ratio of the corresponding sides and, i.e., cm

25 Similar Triangles 53 Now, measure the other sides of the two triangles and find the ratios of the corresponding sides. What do you find? You will find that 0.8. and are also equal to epeat this activity by constructing several pairs of equiangular triangles. In each case, you will find that the corresponding sides are in proportion. This activity leads us to the criterion for similarity of triangles which is referred to as criteria (ngle-ngle riteria). It is stated as "In two triangles, if the corresponding angles are equal, then their corresponding sides will be in proportion and hence the two triangles are similar". O If two triangles are equiangular, then their corresponding sides are proportional". Now, let us prove this statement logically. Theorem ( similarity riterion) "If two triangles are equiangular, then their corresponding sides are proportional". ata : G In and H (i) = (ii) = To prove: onstruction: Mark points 'G' and 'H' on and such that (i) G = and (ii) H = Join G and H roof: Statement eason compare GH and, G = [ onstruction ] GH = [ ata] H = [ onstruction] GH [ SS ] GH = [ T] ut = [ ata]

26 54 UNIT-10 GH = [ xiom-1] GH [ If corresponding. angles are equal then lines are.] in G GH H [ third corollary to Thales theorem] Hence [ GH ] We know that, if two angles of a triangle are equal to corresponding two angles of another triangle, then by the angle sum property of a triangle their thrid angles will also be equal. Therefore, similarity criterion can also be stated as follows: "If two angles of one triangle are equal to corresponding two angles of another triangle, then the two triangles are similar. This is reffered to as the similarity criterion for two triangles. Now, let us study some applications of similar triangles. 1. pplication of criteria to derive mirror formula. Let be the object kept beyond ''. Its image will be '' formed between and. Nature of image will be inverted, diminished and real. M N ompare and I I = I I = 90 0 = [ Vertically opposite angles] I I [ quiangular triangles] I I I I [ criteria]

27 Similar Triangles 55 f I I v f I... (i) ompare and I I = I I = 90 0 = = I I [ Vertically opposite angles] I I [ quiangular triangles] I I I I [ criteria] u f I I f v... (ii) compare (i) and (ii), Since =, we get f u f v f f v uv uf vf + f uv uf = vf vf uv uf = vf = f vf ivide the equation by (uvf) (uv uf = vf) f v u f u v uvf ILLUSTTIV XMLS xample 1: In, L, M and N are the altitudes which concurr at 'O'. rove that (i) M N Sol. ata : In, L, M and N are the altitudes. To prove : (i) M N (ii) N L M (ii).. = 1 N L M N L M.. 1 N L M N O M roof : In M and N M N 90 ( ata) L

28 56 UNIT-10 M = N ( common angle) M N ( criteria) ly L N and M L can be proved M N L N M L N L M M N L N M L N M L N L M 1 N L M or N L M = N L M xample : In the trapezium, (i) (ii). rove that = ata: In the To rove: roof: (i) (ii) = ompare and = [ II and lternate ngles] = [ quiangular triangles] [ ] ut, [ ata] [ ] [ xiom 1]

29 Similar Triangles 57 = = 1 = xample 3: rahma Gupta's theorem (68..) prove that, "The rectangle contained by any two sides of a triangle, is equal to rectangle contained by altitude drawn to the third side and the circum diameter." ata: In, X 'O' is the circumcenter of, is the diameter To prove:. = X. onstruction : Join and roof: ompare X and X 90 [ angle in a semicircle is a right angle] X [ angles in the same segment] X ~ [ quiangular triangles] X [ criteria]. = X. XIS 10.4 iders based on similarity criteria. 1. and are two right angled triangles with common hypotenuse. The sides and intersect at '' rove that. =.

30 58 UNIT-10. In the figure = UTS = 90 0 and TS rove that TUS S U 3. '' is a point on such that rove that T 4. If the diagonals of a quadrilateral divide each other proportionally, then prove that the quadrilateral is a trapezium. 5. is a rhombus. '' is any point on, is joined and produced to meet produced at. rove that The diagonal of a gm intersects at ''. '' is any point on. rove that. =. 7. In the adjoining figure 90 and M 90 rove that (i) M (ii) M 8. In the, O = 3x 19; O = x 5 O = x 3; O = 3 ind 'x' 9. In the trapezium and =, 3 4 rove that If the mid points of three sides of a triangle are joined in an order, then prove that the four triangles so formed are similar to each other and to the original triangle. x 3 x 3 19 O M G 5 x 3

31 Similar Triangles 59 Side-Side-Side (SSS) similarity criterion for two triangles You have seen in the previous theorem that, if the three angles () of one triangle are equal to the corresponding three angles () of another triangle, then their corresponding sides (SSS) are proportional. Let us state the converse of this statement. "If the three sides (SSS) of a triangle are proportional to the corresponding three sides (SSS) of another triangle, then their corresponding angles () are equal". In and, if then = = = i.e., and are equiangular and hence similar. Observe: The following triangles In and ; cm 1 cm 5 cm.5 cm ut the two triangles and are not equiangular (one is obtuse angled and another is an acute angled and hence they are not similar.) rom the above example we can say that if two sides of one triangle are proportional to two sides of another triangle then those two triangles need not be equiangular and hence they are not similar. We can conclude that, : or two triangles to be similar, the fundamental criteria is that all the three sides of one triangle must be proportional to all the three sides of another triangle. Note: iscuss the proof for SSS criteria in groups Side ngle Side (SS) similarity criterion for two triangles. If two sides of one triangle are proportional to two sides of another triangle and the angles formed by those sides are equal, then the two triangles are equiangular and therefore they are similar.

32 60 UNIT-10 ata : In N & IT 1. N IT T. N = IT then, N = TI N = TI Hence N IT 8 cm 4 cm cm N T.5 cm I XIS In which of the following cases the pairs of triangles are similar? Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form. W X Y L X ig 4 M Z ig 1 H 4 U ig 10 G T 6 6 J 4 K L M 5 V 6.9 cm cm O ig 5 1 4cm 5cm 15 cm ig 3 7 M 5 ig cm 1 L G H 7.5 M O 4 4 Z Y ig T 4 ig 6 G J ig 9 I L H

33 Similar Triangles 61 Theorem:- In a right angled triangle, the perpendicular to the hypotenuse from the right angled vertex, divides the original triangle into two right angled triangles, each of which is similar to the original triangle. ata:- In, (i) = 90. (ii). To rove:- Statement roof:- ompare, and eason (i) = = 90 ata (ii) = ommon angle (iii) = Sum of 3 angles of a is (1) quiangular triangles ompare and (i) = = 90 ata (ii) = ommon angle (iii) = Sum of 3 angles of a is () quiangular triangles Now, we have (from 1 and ) Note:- In the third step to prove we have used the results of first two steps and axiom 1. lternatively without all these we can show that they are similar by comporing the angles and show that they are equiangular and hence similar. iscuss in groups and the proof separately.

34 6 UNIT-10 OOLLIS orollary - 1 orollary - orollary - 3 [ quiangular] [ criteria] =. Now recall the definition of gemetric mean (G.M) rom the above equation, we can say that is the geometric mean of and. [ quiangular] [ criteria] =. y the definition of geometric mean (G.M), we can say that is the geometric mean of and. Note:- The converse for the above three corollaries exists and are true. So discuss in groups, the proof for each and write them separately. [ quiangular] [ criteria] =. gain by the definition of geometric mean (G.M) we can say that is the geometric mean of.. xample 1: In, 90 and is the altitude. If = 5, = 4. Then find and. Sol. =. [ orollary]? = 4 = = 5 4 = 1 =. = 5 1 = 5 xample : In, 90 and M is the altitude. If M = 16M rove that = 4 Sol. =.M =. 16 M = 4. M =.M [ orollary] [ orollary] 5 M 4? =. M ut = 4. M = 4. M

35 Similar Triangles 63 XIS In, = 90, (a) If = 8cm, = 4cm, find (b) If = 5.7cm, = 3.8cm, = 5.4cm, find (c) If = 75cm, = 1m, = 1.5m, find. In, = 90,, x y = 4cm, = 5cm, ind x and y. 3. In, = 90, S. 4 c 5 S a p If = a, = b, = c and S = p, show that pc = ab. b 4. In, = 90,. If = 4. rove that =. 5. In, = 90, M (a) M = x +, M = x + 7, M = x, find x. (b) M = 8x, M = x, then find M and. reas of similar Triangles So far we have studied the two fundamental properties regarding angles and sides of similar triangles. You may now be very eager to know the relationship between the areas of similar triangles. To understand the relation, conduct the following activitiy. onsider two similar triangles and, with the following measurments. M In and, cm 6 cm 3.4 cm 3.8 cm 50 M.5 cm 1.7 cm 1.9 cm cm N Now let us consider the ratio between their areas. Let M and N.

36 64 UNIT-10 M = 3.4cm, N = 1.7 cm. rea of Δ ½ M ½ rea of Δ ½ N ½ This means that area of is four times the area of. epeat this activity for two right angled triangles and two obtuse angled triangles which are similar. What is your inference? iscuss in groups. We can generalize this in the following way : "The areas of similar triangles are proportional to squares on the corresponding sides." Theorem : "The areas of similar triangles are proportional to the squares of the corresponding sides" 4 ata: L M To prove: onst.: rea of rea of raw L and M roof: Statement eason ompare L and M L = M L = M = 90 0 [ ata] [ construction] L M [ quiangular] L M [ - criteria] but = [ ata]

37 Similar Triangles 65 L M [ Transitive property] rea of 1 L rea of M 1 1 rea of Δ= b h Now, ar ( ) L ar ( ) M = L M = L M is proved = ar (Δ) = ar (Δ) rom data, = = ar (Δ) = = = ar (Δ) ILLUSTTIV XMLS 1. In, XY and XY divides the triangle into two parts of equal areas. ind X ata: In To find: (i) XY (ii) = XY X X Y Solution: XY [ XY ] ar( ) ar( XY) X [ Thales theorem]

38 66 UNIT-10 1 X 1 X 1 X 1 X 1 X X Important results : 1. X 1. X 3. X 1 X 1. "If the area of two similar triangles are equal, then they are congruent"- rove. ata: * * ar ( ) = ar ( ) To rove: roof: ar (Δ) ar (Δ) [ Theorem] 1 [ areas are equal] by data] = = = = = = [ SSS] 3., and are the mid points of sides of., and are the mid points of sides of. This process of marking the midpoints and forming a new triangle is continued. How are the areas of triangles so formed related? Investigate and draw inference. We know that

39 Similar Triangles 67 ar (Δ) ar (Δ) = 1 4 ar (Δ) ar (Δ) = 1 16 So the areas are1, 1, These are in G. with r = 1 4 Know this! reas of similar triangles are proportional to 1. Squares on their corresponding sides.. Squares on their corresponding altitudes. 3. Squares on their corresponding medians. 4. Squares on their corresponding circum-radii. 5. Squares on their corresponding angular bisectors. 6. Squares on their corresponding in-radii. Note : The converse of the above six statements also holds good. XIS 10.7 iders based on areas of similar triangles. 1. and are on the same base O rove that ar (Δ) O ar (Δ) O. and are two equilateral triangles and =. ind the ratio between areas of and. 3. Two isosceles triangles are having equal vertical angles and their areas are in the ratio 9 : 16. ind the ratio of their corresponding altitudes. 4. The corresponding altitudes of two similar triangles are 3cm and 5cm respectively. ind the ratio between their areas

40 68 UNIT In the trapezium,, = and ar ( O) = 84cm, find the area of O 6. In the above figure find the ratios between areas of O and O, if = rove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. Similar Triangles Similar igures Similar traingles riteria for similarity of triangles asic roportionality Theorem (T) onverse of T orollary 1 riteria Logical proof orollary onverse of criteria- SSS criteria SS riteria Theorem related to ight angled triangle orollary 3 Numerical problems and riders reas of similar triangles Logical proof Numerical problems and riders NSWS XIS ] (i) x = 4, y = 9 (ii) x =.64, y = 0.96 (iii) a = 9, b = 3 6] (i) 3:1 (ii) 15cm (iii) 4cm (iv) 4cm (v) 4cm 7] 15cm, 1cm 8].6cm 10] 88m 11] ft XIS 10. ] = 10.5cm, = 7.5cm 3].1cm 4] (i) 0cm (ii) 6cm (iii) x = 1 5] 7.5cm 8] 6cm 9] K = 1cm, = 10cm 10] 8 ft XIS ] (a) 16cm (b) 6.6 cm (c) 0.6m ] x = 6cm, y = 5 cm 5] (a) 4 3 XIS 10.7 ] 4:1 3] 3:4 4] 9:5 5] 1 cm 6] 9:1 (b) 4 x,4x 5