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Downloaded from https:// www.studiestoday.com, , , , , , , , ! x , side of the triangle can be taken as its base and the corresponding height means : the, € ngth of perpendicular to this side from the opposite vertex., , aan fe A (i) ae, , , , , , , , , , , , , , , , , , , , , , , , Q, RAJ, B P Cc B o B Cc, If BC is taken as base If AC is taken as base If AB is taken as base, 1, area = 5 x BC x AP area= 5 x AC xBQ area= xABXxCR, 1s, 2xarea, - Also area = ~ xbasexheight > (i) base = ~~, ; 2 9 (i) height, 2xarea, iyeheignt = -....., (ily helg base, , , , , , Example 1 :, , Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm. Also, find the length, of altitude corresponding to the largest side of the triangle., , Solution :, Leta=9cm,b=12cmandc=15cm, , —atb+c _ 94+12+15 = a0} =, os Se aia co 5 cm = 18cm, , Area of triangle = yS(s—a)(s—b)(s—°¢), , = ,{18(18 — 9) (18 — 12) (18-15), , , , , , , , , , , , = J18x9x6x3 = /2916 = 54cm? (Ans.), Also, area of triangle = . base x corresponding altitude, 54 = 3 x15xh [Taking largest side as the base], =“ h= Saxe cm = 7-2.em (Ans.), Example 2 :, Find the area of an equilateral triangle, whose one side is a cm., Solution :, gx 4 +b+ a ; 1S “3 [Sides of an equilateral triangle are equal], Area = Js(s —a)(s—b)(s—c), 3a (3a _ ,)(3a _ )(3a + (F a)($ a)($ a), pmsala va, a . axa AIS (5-2, = > *<D*%o*%d = 2x2 3 = oon (Ans.), i * 309 a, , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, , Example 3 :, , The base of an isosceles triangle is 12 cm and its perimeter A, is 32 cm. Find the area of the triangle., , Solution : xom, , Let each of the two equal sides of the given, isosceles triangle be x cm., , Since, perimeter of the triangle is 32 cm B tS ont C4, , — XX li = G2 axel, , Hence, the sides of the given isosceles triangle are 10 cm, 10 cm and 12 cm, , Let a = 10cm, b=10 cm andc= 12 em, , atb+c 10+10+12, 2 2, , /s(s —a)(s—b)(s—c), , = 16(16—10)(16—10)(16—12) cm?, , , , s cm = 16 cm, , Area of the A, , , , Vi6x6x6x4 cm2=4x 6 x 2 cm2 = 48 cm? (Ans.), , Alternative method :, Draw AD perpendicular to base BC of the given triangle., , Since, the perpendicular from the vertex of an isosceles iOcm, triangle to its base bisects the base, therefore, , BC 12, BD =CD= ae = > cm =6cm., In right-angled triangle ABD, agate isan, AD? + BD? = AB?, , =) AD? + 62 = 102, , 10cm, , = AD? = 100-36 =64 and AD= /64 cm=8cm, Now the base BC of the given triangle is 12 cm and its height AD is 8 cm, , 1, Area of the A = > x base x height, , 1, = 5 x 120m x 8 cm =48 cm? . (Ans.), , TEST YOURSELF, , , , , , , ee ee, , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, , re a. i area of an equilateral triangle, with side a cm is numerically equal to its perimeter,, arene VUNG. tarts tccescactess es te eee ee = ct, , , , 1 A, > BSG ep psi asctenc hepseveencaite ss ;, , E, Me Nero raya iv ex sng such dnkuiv ache and, Beira ac et EA G ie ase. ss vawuneectoccennts, , 5. In the given figure, , , , , , , , , , , , , , , , HL) HII pcateceensictes. AVIA ea sariss cece., (b) area Of A ABC = ......sesssecsesssessees = cereeneentenntenneensees, _ - EXERCISE 32 (A), 1. Find the area of of a triangle; whose sides | 6. The base and the height of a triangle are in, are : the ratio 5: 3. If the area of the triangle is, (i) 10cm, 24 cm and 26 cm 67-5 m2; find its base and height., (ii) 18mm, 24 mm and 30 mm 7. The area of an equilateral triangle is, (iii) 21m, 28 mand 35m 144./3 cm?; find its perimeter., 2. Two sides of a triangle are 6 cm and 8 cm. If f : ;, height of the triangle corresponding to 6 cm oe I ela alte, side is 4 cm; find : ae x (side)? and its perimeter = 3 x side, , , , , , , , (i) area of the triangle, , (ii) height of the triangle corresponding to 8. The area of an equilateral triangle is numeri, cally equal to its perimeter. Find its perimeter, , 8 cm side. ’, ; ' correct to 2 decimal places., 3. The sides of a triangle are 16 cm, 12 cm and | g A field is in the shape of a quadrilateral ABCD, 20 cm. Find : in which side AB = 18 m, side AD = 24 m,, (i) area of the triangle side BC = 40 m, DC = 50 m and angle, (ii) height of the triangle, corresponding to A = 90°. Find the area of the field., the largest side 10. The lengths of the sides of a triangle are in, (iii) height of the triangle, corresponding to the ratio 4 : 5 : 3 and its perimeter is 96 cm., the smallest side. Find its area., , 11. One of the equal sides of an isosceles triangle, is 13 cm and its perimeter is 50 cm. Find the, area of the triangle., , 12. The altitude and the base of a triangular field, are in the ratio 6 : 5. If its cost is ¥ 49,57,200, at the rate of = 36,720 per hectare and, , 4. Two sides of a triangle are 6-4 m and 4-8 m., If height of the triangle corresponding to, 4-8 m side is 6 m; find :, (i) area of the triangle;, (ii) height of the triangle corresponding to, , , , , , Gt munice 1 hectare = 10,000 sq. m, find (in metre), 5. The base and the height of a triangle are in dimensions of the field., the ratio 4 : 5. If the area of the triangle is, 40 m2: find its base and height. Area of the given triangular field, saat : Cost of the field _ 49,57,200, 4x m and height be 5x m. 5 aes ee720 “hectare, Ria es = 135 x 10,000 sq. m = 1350000 m2., , , , , , , , , , , , , , , , , , Ss 311, , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, , 13. Find the area of the right-angled triangle with (i) the length of AC. Caq, hypotenuse 40 cm and one of the other two (ii) the are a of A ABC. D, SINGS En ats. ee fa ey aia (iii) the length of BD, E, 14. Use the information given in the adjoining fig- correct to one Ke, ure to find : decimal place., , , , PERIMETER AND AREA OF RECTANGLES, , 1. Perimeter = length of boundary b, 21 + 2b = 2(/ + b), , , , , , , , , , 2: Area = length x breadth, I, =x 0, 3. Since, d? = /2 + b? [Applying Pythagoras Theorem], , . Diagonal (d) = v/2 +b?, 32.4| PERIMETER AND AREA OF SQUARES, , 1. Perimeter = 4a=4~x side, 2. Area = ax a=a? = (side)?, , Va?+a2 = 2a? =a/2 =side /2, , , , , , , , , , , , , , , , 3. Diagonal (d), , Example 4 :, The perimeter of a rectangle is 28 cm and its length is 8 cm. Find its :, (i) breadth (ii) area (iii) diagonal, Solution :, (i) Since, perimeter = 2(/ + b), a 28 = 2(8+b) => b=6cm (Ans.), (ii) Area = 1xb=8x 6 cm? = 48 cm? (Ans.), (iii) Diagonal (d) = Ji? +b? = 8? +62 =10cm (Ans.), Example 5:, The area of a rectangle is 5-4 m2. If its breadth is 1-5 m; find its :, (i) length (ii) perimeter, Solution :, (i) Area = [xb E, => 54=1x1-50rl = o4 m=3-6m (Ans.), (ii) Perimeter = 2(/ + b) = 2 (3-6 + 1-5) m=10-2 cm (Ans.), Example 6:, The perimeter of a square is 28 cm. Find its :, (i) one side (ii) area (iii) diagonal ., , , , , hate 312 aa, , Downloaded from https:// www.studiestoday.com