4 4. Heron s Formula : If a, b, c denote the lengths of the sides of a triangle, then its Area, ILLUSTRATIVE EXAMPLES

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1 CHAPTER Points to Reeber :. Area of right triangle Base height 3. Area of an Equilateral triangle (side) a HERON S FORMULA 3. Area of a isosceles triangle b a, where, a is base and b represents equal sides.. Heron s Forula : If a, b, c denote the lengths of the sides of a triangle, then its Area, A s( s a)( s b)( s c) a b c where, s 5. Area of a quadrilateral can be calculated by dividing the quadrilateral into triangles and using heron s forula for calculating area of each triangle. Exaple. Exaple. ILLUSTRATIVE EXAMPLES Find the area of a triangle whose sides are 3, and 5 respectively. Let a = 3, b =, c = 5 a b c 3 5 s Area s( s a)( s b)( s c) ( 3)()(5) sq. 876 sq sq. = 3 7 sq.. = 8 sq.. Area of triangle = 8 sq.. Ans. A traffic signal board, indicating SCHOOL AHEAD is an equilateral triangle with side a. Find the area of the signal board, using heron s forula if it perieter is 80. NCERT Given perieter of equilateral triangle = 80. Side of an equilateal triangle is a, then its perieter is 3a. 3a = 80 a = 60 Now, a a a s 90. Using heron s forula, Area s( s a)( s a)( s a) 90(90 60)(90 60)(90 60) MATHEMATICS IX HERON S FORMULA 35 Ans.

2 Exaple 3. Exaple. A triangular park ABC has sides 0, 80 and 50 (see figure). A gardener Dhania has to put a fence all around it and also plant grass inside. How uch area does she need to plant? Find the cost of fencing it with barbed wire at the rate of Rs 0 per etre leaving a space 3 wide for a gate on one side. (i) For Area of the park : Let a = 0, b = 80, c = 50 a b c s 5 Area s( s a)( s b)( s c) 5 (5 0)(5 80)(5 50) Also, Perieter of the park = AB + BC + AC = 50 length of wire needed for fencing = 50 3 (to left for gate) = 7 and cost of fencing = 7 0 Rs = Rs. 90 Ans. Triangular side walls of a flyover have been used for advertiseents. The sides of the walls are, and 0 (see figure). The advertiseents yield an earning of Rs per per year. A copany hired one of its walls for 3 onths, how uch rent did it pay? NCERT 36 HERON S FORMULA MATHEMATICS IX

3 Exaple 5. We will calculate area of triangular wall here, a =, b =, c = 0 a b c 0 6 s 3 Area s( s a)( s b)( s c) 3 (3 ) (3 ) (3 0) Now, Rent charges = Rs per per year Rent charged fro copany for 3 onths 3 Rs = Rs. 6,50,000 Ans. There is a slide in a park. One of the its side walls has been painted, in colour with a essage KEEP THE PARK GREEN AND CLEAN, (see figure). If the sides of the wall are 5, and 6, find the area painted in colour. NCERT here, a = 5, b =, c = 6 Exaple 6. a b c s 6 Area s( s a)( s b)( s c) 6(6 5)(6 )(6 6) Area painted in a colour = Area of side wall = 0 Ans. An isosceles triangle has perieter 30 and each of the equal sides is. Find the area of the triangle. NCERT here, a = b =. also, a + b + c = c = 30 c = 6 MATHEMATICS IX HERON S FORMULA 37

4 a b c 30 s 5 Area of the triangle s( s a)( s b)( s c) 5(5 )(5 )(5 6) HERON S FORMULA MATHEMATICS IX Ans. Exaple 7. The perieter of a triangle is 50 and its sides are in the ratio of 3 : : 5. Find the area of the triangle. Let the sides of the triangle by 3 x, x and 5x. Given, perieter of triangle = 50 3 x + x + 5 x = 50 30x = 50 x = 5 Sides are 3 5, 5 and 5 5 i.e. 95, 80 and 75. Let a = 95, b = 80, c = 75 Exaple 8. a b c s 5 Area s( s a)( s b)( s c) 5(5 95)(580)(5 75) sq sq sq.. = sq.. = 6750 sq. Ans. A park, in the shape of a quadrilateral ABCD, has C = 90, AB = 9, BC =, CD = 5 and AD = 8. How uch area does it occupy? NCERT Area of ΔBCD BC CD 5 Now, In BCD, BD = BC + CD BD 5 5 For ABD, let a = 3, b = 8, c = 9. a b c Now, s 5 Area of ΔABD s( s a)( s b)( s c) (5 3)(5 8)(5 9)

5 Exaple (approx.) MATHEMATICS IX HERON S FORMULA 39 = 35. (approx.) Required Area = ar (ABD) + ar (BCD) = = 65. (approx.) Ans. A parallelogra, the length of whose sides are 60 and 5 has one diagonal 65 long. Find the area of the parallelogra. Let ABCD be the given parallelogra. Area of parallelogra ABCD = area of ABC + area of ACD = area of ABC Let, a = 60, b = 65, c = s 75 area of ABC s( s a)( s b)( s c) 75(75 60)(75 65)(75 5) = = 750 Area of parallelogra ABCD = area of ABC = 750 = 500 Ans. Exaple 0. A triangle and a parallelogra have the sae base and the sae area. If the sides of the triangle are 6, 8 and 30, and the parallelogra stands on the base 8, find the height of the parallelogra. NCERT For the triangle ; Let a = 6, b = 8, c = 30 a b c then, s Now, Area of triangle s( s a)( s b)( s c) ( 6)( 8)( 30) = 3 7 = 336 For the parallelogra : Area = base height Area height ( Area of parallelogra = Area of triangle) base 336. Ans. 8

6 Exaple. Find the area of a quadrilateral ABCD, in which B = 90, AB = 5, BC = 36, CA = and DA =. Join A to C. In ABC, B = 90 Area of ABC 365 = 70 also, In ABC, AC = AB + BC = = = 5 AC 5 39 Now, For area of ACD, we will use hero s forula. let, a = 39, b =, c =. a b c 39 s Area of ACD s( s a)( s b)( s c) ( 39)( )( ) 0 HERON S FORMULA MATHEMATICS IX Area of ABCD = = 88.3 Ans. Exaple. Radha ade a picture of an aeroplane with coloured paper as shown in the figure. Find the total area of the paper used. NCERT Area I = area of isosceles triangle with a = and b = 5 a b a 5 ()

7 99.5 (approx) Area II = area of rectangle = length breadth = 6.5 = 6.5 Area III = Area of Trapeziu = 3 area of equilateral triangle with a = = 3 () =.99.3 (approx) Area of IV and V Total area of paper used = area I + area II + area III + area IV + area V = = 9.3 (approx) Ans. Exaple 3. An Ubrella is ade by stitching 0 triangular pieces of cloth of two different colours (see figure), each piece easuring 0, 50 and 50. How uch cloth of each colour is required for the ubrella? NCERT In one triangular piece, a = 0, b = 50, c = 50. Now, s 60 Area s( s a)( s b)( s c) 60 (60 0)(60 50)(60 50) MATHEMATICS IX HERON S FORMULA 6 Area of 5, Ist coloured triangles = = and area of 5, IInd coloured triangles = = Exaple. A kite in the shape of a square with a diagonal 3 and an isosceles triangle of base 8 and sides 6 each is to be ade of three different shades as shown in figure. How uch paper of each shade has been used in it. NCERT Each diagonal of square = 3 CO 3 6 Area of each shaded portion I and II base height And, ara of IIIrd portion with sides 6, 6 and 8. a 8 b a (6) (8)

8 Ans. Exaple 5. A floral design on a floor is ade up of 6 tiles which are triangular, the sides of the triangle being 9, 8 and 35 (see figure). Find the cost of polishing the tiles at the rate of 50 p per. NCERT For each triangular tile, Let a = 9, b = 8, c = 35 a b c s 36 Area of each triangular tile s( s a)( s b)( s c) Total area of floral design Cost of polishing the 6 triangular tiles HERON S FORMULA MATHEMATICS IX 36 (36 9)(36 8)(36 35) Rs Rs Rs Ans. Exaple 6. Find the area of a trapeziu ABCD in which AB DC, AB = 77, BC = 5, CD = 60 and DA = 6. Draw DE BC and DF AB the, DE = BC= 5. AE = AB EB = AB DC = = 7. In DAE, Let a = 7, b = 5, c = 6 a b c s 3 Area of DAE s( s a) ( s b) ( s c) Also, area of ΔDAE AE DF 3 (3 7)(3 5)(3 6) () 7 DF...() fro () and (), we get 7 DF 0 0 DF 7 area of trapeziu ABCD (AB DC) DF

9 (77 60) = 6 Ans. PRACTICE EXERCISE. Find the area of the triangle whose sides are : (i) 9, and 5 (ii) 50, 0 and (iii) 6, and 5 (iv) 6, 8, 30. Find area of a equilateral triangle of side An isosceles triangle has a perieter 30 and each of the equal sides is. Find the area of the triangle.. Calculate the area of the shaded portion, in the given figure. 5. The sides of a triangle are in the ratio 3::5 and its perieter is 0. Find its area. 6. If the area of an equilateral triangle is 9 3, find its perieter. 7. Two sides of a triangular field are 85 and 5 in length and its perieter is 3. Find (i) The area of the field. (ii) the length of perpendicular fro the opposite vertex on the side easuring ABC is a right triangle right angled at A with AB = 6, AC = 8. A circle with centre O and radius r is inscribed in the triangle. Find r. 9. The perieter of a triangular field is 0. If two of its sides are 78 and 50, find the length of the perpendicular on the side of length 50 fro the opposite vertex. Calculate also the cost of watering it at Rs per 00 sq.. 0. A triangle has sides 35, 5 and 6 long. Find its area. Also, find the sallest altitude.. Find the area of a quadrilateral ABCD in which AB = 3, BC =, CD =, DA = 5 and AC = 5.. The area of a rhobus is 7. If one of the diagonals is 8 long, find the length of the other diagonal. 3. The diagonal of a four sided field is 0. The perpendiculars fro the opposite vertices on this diagonal are 0 and. Find the area of the field.. A field is in the shape of a trapeziu whose parallel sides are and 5. The non-parallel sides are 6 and 30. Find the area of the field. MATHEMATICS IX HERON S FORMULA 3

10 5. Find area of the following polygons : MM : 0 General Instructions : (i) Each Question carry arks. PRACTICE TEST (ii) Tie : hour. The perieter of a right triangle is 50 and its sides are in the ratio 3 : : 5. Find the area of the triangle.. Find the area of a rhobus, one of whose sides is 5 and one of whose diagonal is In a ABC, AB =, BC = 9 and AC = 7. Find the length of the perpendicular fro A to BC.. A triangle and a parallelogra have the sae base and the sae area. If the sides of the triangle are 6, 8 and 30, and the parallelogra stands on the base 8, find the height of the parallelogra. 5. A field is in the shape of a trapeziu whose parallel sides are 55, 0 and non-parallel sides are 0, 5. Find the area of the field. ANSWERS OF PRACTICE EXERCISE. (i) 5 (ii) 8969 (iii) 0 (iv) (i) 77 (ii) , Rs , (i) 9. (ii) ANSWERS OF PRACTICE TEST HERON S FORMULA MATHEMATICS IX

2 a. 3 (60 cm) cm cm 4

2 a. 3 (60 cm) cm cm 4 Class IX - NCERT Maths Exercise (1.1) Question 1: A traffic signal board, indicating SCHOOL AHEAD, is an equilateral triangle with side a. Find the area of the signal board, using Heron s formula. If its