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1 Exercise Find the area of a triangle whose sides are respectively 150 cm, 10 cm and 00 cm. The triangle whose sides are a = 150 cm b = 10 cm c = 00 cm The area of a triangle = s(s a)(s b)(s c) Here 1s = semi perimeter of triangle s = a + b + c S = a+b+c = = 35cm area of triangle = s(s a)(s b)(s c) = 35(35 150)(35 00)(35 10) = 35(85)(35)(115)cm = cm. Find the area of a triangle whose sides are 9 cm, 1 cm and 15 cm. The triangle whose sides are a = 9cm, b = 1 cm and c = 15 cm The area of a triangle = s(s a)(s b)(s c) Here 1s = semi-perimeter of a triangle s = a + b + c S = a+b+c = = 36 = 18 cm area of a triangle = s(s a)(s b)(s c) = 18(18 9)(18 1)(18 15) = 18(9)(6)(3) = 18 cm 3 cm 54cm = 54 cm. 3. Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 4cm. 1 11cm

2 4. In a ABC, AB = 15 cm, BC = 13 cm and AC = 14 cm. Find the area of ABC and hence its altitude on AC. The triangle sides are Let a = AB = 15 cm, BC = 13cm = b. c = AC = 14 cm say. Now, s = a + b + c S = 1 (a + b + c) s = ( ) cm s = 1 cm area of a triangle = s(s a)(s b)(s c) = 1(1 15)(1 14)(1 13) cm = cm = 84 cm Let BE be perpendicular ( er ) to AC Now, area of triangle = 84 cm 1 BE AC = 84 BE = 84 AC BE = = 1cm Length of altitude on AC is 1 cm. 5. The perimeter of a triangular field is 540 m and its sides are in the ratio 5 : 17 : 1. Find the area of the triangle. The sides of a triangle are in the ratio 5 : 17 : 1

3 Let the sides of a triangle are a = 5x, b = 17 x and c = 1x say. Perimeter = 5 = a + b + c = 540 cm 5x + 17x + 1x = 540 cm 54x = 540cm x = x = 10 cm The sides of a triangle are a = 50 cm, b = 170 cm and c = 10 cm Now, Semi perimeter s = a+b+c = 540 = 70 cm The area of the triangle = s(s a)(s b)(s c) = 70(70 50)(70 170)(70 10) = 70(0)(100)(150) = (9000)(9000) = 9000 cm The area of triangle = 900 cm. 6. The perimeter of a triangle is 300 m. If its sides are in the ratio 3 : 5 : 7. Find the area of the triangle. Given that The perimeter of a triangle = 300 m The sides of a triangle in the ratio 3 : 5 : 7 Let 3x, 5x, 7x be the sides of the triangle Perimeter s = a + b + c 3x + 5x + 17x = x = 300 x = 0m The triangle sides are a = 3x = 3 (0)m = 60 m b = 5x = 5(0) m = 100m c = 7x = 140 m Semi perimeter s = a+b+c = 300 m = 150m The area of the triangle = s(s a)(s b)(s c)

4 = 150(150 60)( )( ) = = cm le Area = cm 7. The perimeter of a triangular field is 40 dm. If two of its sides are 78 dm and 50 dm, find the length of the perpendicular on the side of length 50 dm from the opposite vertex. ABC be the triangle, Here a = 78 dm = AB, BC = b = 50 dm Now, perimeter = 40 dm AB + BC + CA = 40 dm AC = 40 BC AB AC = 11 dm Now, s = AB + BC + CA s = 40 s = 10 dm Area of ABC = s(s a)(s b)(s c) by heron s formula = 10(10 78)(10 50)(10 11) = = 1680 dm Let AD be perpendicualar on BC Area of ABC = 1 AD BC (area of triangle = 1 b h) = 1 AD BC = 1680 AD = = 67. dm

5 8. A triangle has sides 35 cm, 54 cm and 61 cm long. Find its area. Also, find the smallest of its altitudes. The sides of a triangle are a = 35 cm, b = 54 cm and c = 61 cm Now, perimeter a + b + c = 5 S = 1 ( ) s = 75 cm By using heron s formula Area of triangle = s(s a)(s b)(s c) = 75(75 35)(75 54)(75 61) = 75(40)(1)(14) = cm The altitude will be a smallest when the side corresponding to it is longest Here, longest side is 61 cm [ Area of le = 1 b h] = 1 base height 1 h 61 = h = = cm 61 Hence the length of the smallest altitude is cm 9. The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side. Let the sides of a triangle are 3x, 4x and 5x. Now, a = 3x, b = 4x and c = 5x The perimeter s = 144 3x + 4x + 5x = 144 [ a + b + c = s] 1x = 144 x = 1 sides of triangle are a = 3(x) = 36cm b = 4(x) = 48 cm c = 5(x) = 60 cm Now semi perimeter s = 1 (a + b + c) = 1 (144) = 7cm By heron s formulas Area of le = s(s a)(s b)(s c) = 7(7 36)(7 48)(7 60) = 864cm Let l be the altitude corresponding to longest side, 1 60 l = 864

6 l = l = 8.8cm Hence the altitude one corresponding long side = 8.8 cm 10. The perimeter of an isosceles triangle is 4 cm and its baše is ( 3 ) times each of the equal sides. Find the length of each side of the triangle, area of the triangle and the height of the triangle. Let x be the measure of each equal sides Base = 3 x x + x + 3 x = 4 7 x = 4 x = 1 cm Sides are a = x = 1 cm b = x = 1 cm c = x = 3 (1) cm = 18 cm By heron s formulae [ Perimeter = a + b + c = 4 cm] Area of triangle = s(s a)(s b)(s c)cm = 1(9)(9)(1 18) cm = (1)(9)(9)(3)cm = 71.4cm Area of triangle = 71.4 cm 11. Find the area of the shaded region in Fig. Below. Area of shaded region = Area of ABC Area of ADB Now in ADB AB = AD + BD --(i) Given that AD = 1 cm BD = 16 cm Substituting the values of AD and BD in the equation (i), we get

7 AB = AB = AB = 400 AB = 0 cm Area of triangle = 1 AD BD = = 96 cm Now In ABC, S = 1 (AB + BC + CA) = 1 ( ) = 1 (10) = 60 cm By using heron s formula We know that, Area of le ABC = s(s a)(s b)(s c) = 60(60 0)(60 48)(60 5) = 60(40)(1)(8) = 480 cm = Area of shaded region = Area of ABC Area of ADB = (480 96)cm = 384 cm Area of shaded region = 384 cm Exercise Find the area of a quadrilateral ABCD is which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm. For PQR PQ = QR + RP

8 (5 ) = (3) + (4) [ PR = 3 QR = 4 and PQ = 5] So, PQR is a right angled triangle. Right angle at point R. Area of ABC = 1 QR RP = = 6cm For QPS Perimeter = s = AC + CD + DA = ( )cm = 14 cm S = 7 cm By Heron s formulae Area of le s(s a)(s b)(s c) cm Area of le PQS = 7(7 5)(7 4)(7 3)cm = 7 3 cm = 1cm = ( 4.583)cm = cm Area of PQRS = Area of PQR + Area of PQS = ( )cm = cm. The sides of a quadrangular field, taken in order are 6 m, 7 m, 7m are 4 m respectively. The angle contained by the last two sides is a right angle. Find its area. The sides of a quadrilateral field taken order as AB = 6m BC = 7 m CD = 7m and DA = 4 m Diagonal AC is joined Now ADC By applying Pythagoras theorem AC = AD + CD AC = AD + CD AC = 4 + 7

9 AC = 65 = 5 m Now area of ABC S = 1 (AB + BC + CA) = 1 ( ) = = 78 = 39m. By using heron s formula Area ( ABC) = S(S AD)(S_BC)(S CA) = 39(39 6)(39 1)(39 5) = = cm Now for area of ADC S = 1 (AD + CD + AC) = 1 ( ) = 8m By using heron s formula Area of ADC = S(S AD)(S DC)(S CA) = 8(8 4)(8 7)(8 5) = 84m Area of rectangular field ABCD = area of ABC + area of ADC = = 375.8m 3. The sides of a quadrilateral, taken in order are 5, 1, 14 and 15 meters respectively, and the angle contained by the first two sides is a right angle. Find its area. Given that sides of quadrilateral are AB = 5 m, BC = 1 m, CD = 14 m and DA = 15 m AB = 5m, BC = 1m, CD = 14 m and DA = 15 m Join AC Area of ABC = 1 AB BC [ Area of le = 1 (3x + 1)] = = 30 cm In ABC By applying Pythagoras theorem. AC = AB + BC AC = = = 169 = 13 m

10 Now in ADC Let s be the perimeter s = (AD + DC + AC) S = 1 ( ) = 1 4 = 1m By using Heron s formula Area of ADC = S(S AD)(S DC)(S AC) = 1(1 15)(1 14)(1 13) = = 84m Area of quadrilateral ABCD = area of ( ABC) + Area of ( ADC) = = 114 m 4. A park, in the shape of a quadrilateral ABCD, has C = 900, AB = 9 m, BC = 1 m, CD = 5 m and AD = 8 m How much area does it occupy? Given sides of a quadrilaterals are AB = 9, BC = 1, CD = 05, DA = 08 Let us joint BD In BCD applying Pythagoras theorem. BD = BC + CD = (1) + (5) = = 169 BD = 13m Area of BCD = 1 BC CD = [1 1 5] m = 30 m

11 For ABD S = perimeter = 15cm By heron s formula = s(s a)(s b)(s c) = (9+8+13) Area of the triangle = 15(15 9)(15 8)(15 13)m = 15(6)(7)()m = 6 35 m = m Area of park = Area of ABD + ABD + Area of BCD = m = 65.5 m (approximately) 5. Two parallel side of a trapezium are 60 cm and 77 cm and other sides are 5 cm and 6 cm. Find the area of the trapezium. Given that two parallel sides of trapezium are AB = 77 and CD = 60 cm Other sides are BC = 6 m and AD = 5 cm. Join AE and CF Now, DE AB and CF AB DC = EF = 60 cm Let AE = x BF = x = 17 x In ADE, DE = AD AE = 5 x [ Pythagoras theorem] And in BCF, CF = BC BF [ By Pythagoras theorem] CF = 6 (17 x) But DE = CF DE = CF 5 x = 6 (17 x) 5 x = 5 (89 + x 34x) [ (a b) = a ab + b ] 65 x = x + 34x 34x = 38 x = 7 DE = 5 x = 65 7 = 516 = 4cm Area of trapezium = 1 (sum of parallel sides) height = 1 (60 77) 4 = 1644cm

12 6. Find the area of a rhombus whose perimeter is 80 m and one of whose diagonal is 4 m. Given that, Perimeter of rhombus = 80m Perimeter of rhombus = 4 side 4a = 80 a = 0m Let AC = 4 m OA = 1 AC = 1 4 = 1m In AOB OB = AB OA [By using Pythagoras theorem] OB = 0 1 = = 56 = 16 m Also BO = OD [Diagonal of rhombus bisect each other at 90 ] BD = 0B = 16 = 3 m Area of rhombus = = 384m [ Area of rhombus = 1 BD AC] 7. A rhombus sheet, whose perimeter is 3 m and whose one diagonal is 10 m long, is painted on both sides at the rate of Rs 5 per m. Find the cost of painting. Given that, Perimeter of a rhombus = 3 m We know that, Perimeter of rhombus = 4 side 49 = 3m a = 8 m Let AC = 10 = OA = 1 AC = 1 10 = 5m

14 9. Find the area of a quadrilateral ABCD in which AB = 4 cm, BC = 1 cm, CD = 9 cm, DA =34 cm and diagonal BD =0 cm. Given that Sides of a quadrilateral are AB = 4 cm, BC = 1 cm, CD = 9 cm DA = 34 cm and diagonal BD = 0 cm Area of quadrilateral = area of ADB + area of BCD. Now, area of ABD Perimeter of ABD We know that s = AB + BD + DA S = 1 (AB + BD + DA) = 1 ( ) = 96 = 96 = 48 cm Area of ABD = S(S AB)(S BD)(S DA) = 48(48 4)(48 0)(48 34) = 48(14)(6)(8) = 336 cm Also for area of BCD, Perimeter of BCD s = BC + CD + BD S = 1 ( ) = 35 cm By using heron s formulae Area of BCD = s(s bc)(s cd)(s db) = 35(35 1)(35 9)(35 0) = 10 10cm = 10cm Area of quadrilateral ABCD = = 546 cm

15 10. Find the perimeter and area of the quadrilateral ABCD in which AB = 17 cm, AD =9 cm, CD = lcm, ACB = 90 and AC=l5cm. The sides of a quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 1 cm, ACB = 90 and AC = 15 cm Here, By using Pythagoras theorem BC = = 89 5 = 64 = 8cm Now, area of ABC = = 60 cm For area of le ACD, Let a = 15 cm, b = 1 cm and c = 9 cm Therefore, S = = 36 = 18 cm Area of ACD = s(s a)(s b)(s c) = 18(18 15)(18 1)(18 9) = = (18 3) = 54 cm Thus, the area of quadrilateral ABCD = = 114 cm 11. The adjacent sides of a parallelogram ABCD measure 34 cm and 0 cm, and the diagonal AC measures 4 cm. Find the area of the parallelogram. Given that adjacent sides of a parallelogram ABCD measure 34 cm and 0 cm, and the diagonal AC measures 4 cm. Area of parallelogram = Area of ADC + area of ABC [ Diagonal of a parallelogram divides into two congruent triangles] = [Area of ABC] Now for Area of ABC Let s = AB + BC + CA [ Perimeter of ABC] S = 1 (AB + BC + CA) S = 1 ( )

17 Given that the sides of AOB are AO = 4 cm OB = 5 cm BA = 14 cm Area of each equal strips = Area of le AOB Now, for area of AOB Perimeter of AOB Let s = AO + OB + BA s = 1 (AO + OB + BA) = 1 ( ) = 3 cm By using Heron sformulae Area of ( AOB) = s(s ao)(s ob)(s ba) = 3(3 5)(3 5)(3 14) = 3(7)(4)(18) = 168 cm Area of each type of paper needed to make the hand fan = 5 (area of AOB) = = 840 cm 14. A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 13 cm, 14 cm and 15 cm and the parallelogram stands on the base 14 cm, find the height of the parallelogram. The sides of a triangle DCE are DC = 15 cm, CE = 13 cm, ED = 14 cm Let h be the height of parallelogram ABCD Given, Perimeter of DCE

18 s = DC + CE + ED S = 1 ( ) S = 1 (4) S = 1 cm Area of DCE = s(s dc)(s ce)(s ed) = 1(1 15)(1 13)(1 14) = [By heron s formula] = = 84cm Given that Area of le DCE = area of ABCD = Area of parallelogram ABCD = = 84cm 4 h = 84 [ Area of parallelogram = base height] h = 6 cm

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