Class 9 Herons Formula

Transcription

1 ID : in-9-herons-formula [1] Class 9 Herons Formula For more such worksheets visit Answer the questions (1) An umbrella is made by stitching 11 triangular pieces of cloth each piece measuring 4 cm, 13 cm and 13 cm. How much cloth is required for this umbrella. () The base of an isosceles triangle is L cm and its perimeter is X cm. Find the area of the triangle. (3) If in the figure below BECD is a square, BC = 0 cm, CA = 11 cm and BD = 1 cm, find the area of the triangle ABC. (4) The area of a triangle with sides 0 cm, 15 cm and 7 cm is : (5) A carpenter has cut a board in the shape of trapezium. if the parallel sides of the trapezium are 8 cm and 5 cm and non parallel sides are 5 cm and 6 cm, find the area of the board. (6) Find the percentage increase in the area of a triangle if each side is increased by N times. (7) The perimeter of an isosceles triangle is 4 cm and its unequal side is 10 cm. Find the area of the triangle. (8) The area of a trapezium is 56 cm and the height is 4 cm. Find the lengths of its two parallel sides if one side is 4 cm greater than the other. (9) A rhombus has all its internal angles equal. If one of the diagonals is 15 cm, find the length of other diagonal and the area of the rhombus. Copyright 017

2 (10) Find the area of the figure below. Also find the altitude to the base of the triangle. ID : in-9-herons-formula [] (All measurements are in centimeters) (11) The perimeter of a rhombus is 5 cm and one of its diagonals is 4 cm. What is the length of other diagonal. (1) Find the area of a quadrilateral ABCD where AB = 10 cm, BC = 17 cm, CD = 13 cm, AD = 0 cm and AC = 1 cm. (13) The base of an isosceles triangle is 10 cm and perimeter is 36 cm. Find the area of the triangle, if the base is not equal to the other two sides. (14) The perimeter of a triangle is 54 cm. One side of a triangle is cm longer than the smallest side and the third side is 1 cm less than 9 times the smallest side. Find the area of the triangle. Choose correct answer(s) from the given choices (15) Which triangle has the maximum area for a given perimeter : a. Obtuse Angle Triangle b. Isoceles Triangle c. Equilateral Triangle d. Can not be determined 017 Edugain ( All Rights Reserved Many more such worksheets can be generated at Copyright 017

3 Answers ID : in-9-herons-formula [3] (1) 660 cm Following picture shows the triangular piece of cloth, Since all sides of the triangle are known, the area of the triangle can be calculated using Heron's formula. S = (AB + BC + CA)/ = ( )/ = 5 cm The area of the ΔABC = [ S (S - AB) (S - BC) (S - AC) ] = [ 5(5-4) (5-13) (5-13) ] = 60 cm Step According to the question, the umbrella is made by stitching 11 triangular pieces of cloth. The cloth required for this umbrella = 11 Area(ABC) = = 660 cm Copyright 017

4 () cm ID : in-9-herons-formula [4] As per Heron's formula, the area of a triangle with sides a, b and c, and perimeter S = Here S = X Step Here we have an isosceles triangle, so two sides are equal Let's assume a=b, and c=c is the base Also, perimeter X = a + b + c = a + L a = X - L Area = = = (S - X - L ) Substituting S = X in the equation we get Area = = Copyright 017

5 (3) Area : 66 cm ID : in-9-herons-formula [5] If we look at the figure carefully, we notice that, 'BD' is the altitude to the base of the triangle ABC. Step According to the question, the altitude(bd) to the base of the triangle ABC = 1 cm, The base(ca) of the triangle ABC = 11 cm BD CA The area of the ΔABC = 1 11 = = 66 cm (4) 4 cm The area of a triangle with sides a,b, c is given by Heron's formula as Area = Where S is half of the perimeter, i.e S = a + b + c Step Here S = = 4 = 1 Area = = = 4 cm (5) 636 cm Copyright 017

6 ID : in-9-herons-formula [6] Following picture shows the trapezium ABCD, Step Let's draw the line DE parallel to the line BC. We know that the distance between the two parallel lines at every point must be equal. Therefore, EB = DC = 5 cm AE = AB - EB = 8-5 = 3 cm Now, we can see that, this trapezium consists of a triangle ΔADE and a parallelogram BCDE. The area of the triangle ΔADE can be calculated using Heron's formula, since all sides of the triangles are known. S = (AE + DE + AD)/ = ( )/ = 7 cm. The area of the ΔADE = [ S (S - AE) (S - DE) (S - AD) ] = [ 7(7-3) (7-5) (7-6) ] = 36 cm Step 4 The height of the triangle ΔADE and the parallelogram BCDE is equal. Let's assume, the height of the triangle ΔADE be 'h', as shown in the following figure. We know that the area of a triangle = 1/(Base Height), Copyright 017

7 Therefore, the height of the ΔADE = (The area of the ΔADE) Base ID : in-9-herons-formula [7] or h = 36 = 7 cm 3 3 The area of the parallelogram BCDE = EB h = = 600 cm Step 5 The area of the board = Area(ADE) + Area(BCDE) = = 636 cm (6) Consider a triangle QRS with sides a, b and c. Let S = a+b+c Area of triangle QRS = A 1 Step Increasing the side of each side by N times, we get a new triangle XYZ XYZ has sides Na, Nb and Nc By Heron's formula Area of new triangle = Na + Nb + Nc Where S 1 = = N x a+b+c = MS Area of XYZ = = = N x A 1 This means the area increases by Copyright 017

8 (7) 660 cm ID : in-9-herons-formula [8] Here we have an isosceles triangle with perimeter 4 cm and the unequal side of 10 cm. The combined size of the other two sides = 4 cm - 10 cm = 1 cm. Since the triangle is isosceles, the other two sides are equal, and the length of each of them will be: 1 = 61 cm. Step Now we know that the three sides of the triangle are 61, 61 and 10 cm respectively. We can now use the Heron's formula to calculate the area of the triangle: Area of a triangle =, where a, b, and c are its sides and S = Let us first calculate the value of S: S = = 4 = 11 Step 4 a+b+c. Area of the triangle = = 660 cm Copyright 017

9 (8) Lengths : 1 cm and 16 cm. ID : in-9-herons-formula [9] The following picture shows the trapezium ABCD, Let's assume, CD = x cm and CD AB. According to the question, one side of the trapezium is 4 cm greater than the other. Therefore, AB = x + 4, Area(ABCD) = 56 cm, Height of the trapezium ABCD = 4 cm. Step The area of the trapezium ABCD = The height of the trapezium ABCD AB + CD AB + CD = Area(ABCD) The height of the trapezium ABCD x x = 56 4 x = 8-4 x = 4 x = 1 cm. Now, CD = 1 cm, AB = = 16 cm. Thus, the lengths of its two parallel sides are 1 cm and 16 cm respectively. (9) 15 cm, 11.5 cm The key thing to note is that all the internal angles of a rhombus add up to 360 So if all the internal angles are equal, then one internal angle is = 90 This is a square with all sides equal. So the other diagonal is also 15 cm in length The area of a rhombus/square is half the product of the diagonals Area = (15 x 15) = 5 = 11.5 cm Copyright 017

10 (10) Area : 36 cm Altitude : 4 cm ID : in-9-herons-formula [10] The area of the triangle can be calculated using Heron's formula, since all sides of the triangle are known. S = ( )/ = 7 cm. The area of the triangle = [ 7(7-3) (7-5) (7-6) ] = 36 cm Step Let's assume, 'h' is the altitude to the base of the triangle, (The area of the triangle) The altitude to the base of the triangle = The base of the triangle = 36 3 = 4 cm Copyright 017

11 (11) 10 cm ID : in-9-herons-formula [11] One way to solve this is as follows We know that the a) The sides of a rhombus are equal. Therefore one side = 1 4 x 5 = 13 b) A diagonal of a rhombus divides the rhombus into equal triangles c) The area of a rhombus is 1 4 (Diagonal1 x Diagonal) Step Taking one of the two triangles formed by the diagonal with length 4 cm Area (using Heron's formula) = Where S = x = 50 = 5 Area = = 10 (the details of this computation are left the the student) From c) above, Area = 10 = 1 4 (Diagonal1 x Diagonal) = 1 4 (4 x Diagonal) Diagonal = x 10 4 = 10 cm (1) 10 cm Following picture shows the quadrilateral ABCD, Step Let's draw the diagonal AC in the quadrilateral ABCD, Copyright 017

12 ID : in-9-herons-formula [1] Now, we can see that, this quadrilateral consists of two triangles, i.e. ΔABC and ΔACD, the area of each triangle can be calculated using Heron's formula, since all sides of the triangles are known. Step 4 The area of the ΔABC can be calculated using Heron's formula. S = (AB + BC + AC)/ = ( )/ = 4 cm. The area of the ΔABC = [ S (S - AB) (S - BC) (S - AC) ] = [ 4(4-10) (4-17) (4-1) ] = 84 cm Step 5 Similarly, the area of the ΔACD can be calculated using Heron's formula. S = (AC + CD + AD)/ = ( )/ = 7 cm. The area of the ΔACD = [ S (S - AC) (S - CD) (S - AD) ] = [ 7(7-10) (7-17) (7-1) ] = 16 cm Step 6 The area of the quadrilateral ABCD = The area of the ΔABC + The area of the ΔACD = = 10 cm Copyright 017

13 ID : in-9-herons-formula [13] (13) 60 cm As per Heron's formula, the area of a triangle with sides a, b and c, and perimeter S = Step Here we have an isosceles triangle, so two sides are equal Let's assume a=b Area = = We are told that the base (side c) = 10 cm Also, perimeter = S = 36 S = 36 = 18 Also, Perimeter = a + b + c 36 = a + c = a a = = 13 Step 4 Substituting, we get Area = = = 60 cm Copyright 017

14 (14) Area : 36 cm ID : in-9-herons-formula [14] Let's assume the smallest side of the triangle be x cm. Step According to the question, one side of the triangle is cm longer than the smallest side, The length of the side = x + The third side is 1 cm less than 9 times the smallest side, the length of the third side = 9x - 1 Step 4 The perimeter of the triangle is 54 cm, Therefore, x + (x + ) + (9x - 1) = 54 x + x + + 9x - 1 = 54 11x = x = x = 3 Now, x + = 3 + = 5, 9x - 1 = (9 3) - 1 = 6 Step 5 Therefore, all sides of the triangle are 3 cm, 5 cm and 6 cm. Step 6 Following picture shows the triangle, The area of the ΔABC can be calculated using Heron's formula, since all sides of the triangle are known. S = 54/ = 7 cm The area of the ΔABC = [ S (S - AB) (S - BC) (S - CA) ] = [ 7(7-3) (7-5) (7-6) ] = 36 cm (15) c. Equilateral Triangle Copyright 017