Mensuration. Introduction Perimeter and area of plane figures Perimeter and Area of Triangles

Size: px

Start display at page:

Download "Mensuration. Introduction Perimeter and area of plane figures Perimeter and Area of Triangles"

Valerie Parsons
5 years ago
Views:

5 Introduction In previous classes, you have learnt about the perimeter and area of closed plane figures such as triangles, squares, rectangles, parallelograms, trapeziums and circles; the area

1 5 Introduction In previous classes, you have learnt about the perimeter and area of closed plane figures such as triangles, squares, rectangles, parallelograms, trapeziums and circles; the area between two rectangles i.e. area of pathways or borders and area between two concentric circles. You have also learnt the concept of the surface area and volume of cube and cuboid, measurement of surface area and volume of such solids by using basic units. In this chapter, we shall review and strengthen all these. 5. Perimeter and area of plane figures The perimeter of a closed plane figure is the length of its boundary i.e. the sum of lengths of its sides. The unit of measurement of perimeter is the unit of length. The area of a closed plane figure is the measurement of the region (surface) enclosed by its boundary (sides). It is measured in square units i.e. square centimetres (abbreviated cm ) or square metres (abbreviated m ) etc. 5. Perimeter and rea of Triangles (i) rea of a triangle = base height. ny side of the triangle can be taken as its base, then the length of perpendicular (altitude) from the vertex opposite to this side is called its corresponding height. (ii) If is any triangle with sides a, b and c, then perimeter = a + b + c, and area = ss ( a)( s b)( s c) ase Height where s = semi-perimeter = a + b + c (This is known as Heron s formula.). c b a

2 5.. Some special types of triangles (i) Right-angled triangle: If is a triangle in which = 90, then its area = = (product of sides containing right angle). (ii) Equilateral triangle: Let be an equilateral triangle with side a and be the perpendicular from to, then is the mid-point of i.e. = a. a In, = ad = a a = a = area of = a. a = a a = a. Perimeter of = a. (iii) Isosceles triangle: Let be an isosceles triangle with = = a and = b, and be the perpendicular from to, then is mid-point of i.e. = b. In, = ad = = a b = a b a = b a b. area of = = b a b = b a b. Perimeter of = a + b. a b Illustrative Examples Example. alculate the area of a triangle whose sides are cm, 5 cm and cm. Hence, calculate the altitude using the longest side as base. Leave your answer as a fraction. Solution. Since the sides of the triangle are cm, 5 cm and cm. s = cm = 5 cm. rea of triangle = ss ( a)( s b)( s c) = 5( 5 )( 5 5)( 5 ) cm = 5 0 cm = 0 cm. 97

3 The longest side of the triangle is cm, let h cm be the corresponding altitude, then area of triangle = base height 0 = h h = 60. The required altitude of the triangle = Example. The triangular side walls of a flyover have been used for advertisements. The sides of the walls are m, m and 0 m (shown in the adjoining figure). The advertisements yield an earning of ` 5000 per m per year. company hired one of its walls for months. How much rent did it pay? Solution. Here, s = m = m. Using Heron s formula, 8 cm. m 0 m m area of one triangular wall = ss ( a)( s b)( s c) Rent = ` 5000 per m per year. = ( )( )( 0) m = 0 0 m = 0 m \ Rent of one wall for months = ` = ` Example. The perimeter of a triangle is 50 cm. One side of a triangle is cm longer than the smallest side and the third side is 6 cm less than twice the smallest side. Find the area of the triangle. Solution. Let the smallest side of the triangle be x cm, then the other two sides are (x + ) cm and (x 6) cm. Given, perimeter of the triangle = 50 cm x + (x + ) + (x 6) = 50 x = 5 x = \ The lengths of three sides of the triangle are cm, ( + ) cm and ( 6) cm i.e. cm, 7 cm and 0 cm. Here, s = semi-perimeter = 50 Using Heron s formula, cm = 5 cm. area of the triangle = ss ( a)( s b)( s c) = 5( 5 )( 5 7)( 5 0) cm = cm = 0 0 cm. Example. Find the area of a triangle whose perimeter is cm, one side is 9 cm and the difference of the other two sides is cm. Solution. Let the other two sides of the triangle be a cm and b cm, a > b. Then 9 + a + b = a + b = and a b = On solving (i) and (ii), we get a = 8 and b = 5. The sides of the triangle are 9 cm, 8 cm and 5 cm. s = semi-perimeter = cm = cm. 98 Understanding ISE mathematics Ix (i) (ii)

4 area of the triangle = ss ( a)( s b)( s c) = ( 9)( 8)( 5) cm = 6 cm = 6 cm. Example 5. From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. If the lengths of the perpendiculars are cm, 0 cm and 6 cm, find the area of the triangle. Solution. Let be an equilateral triangle with length of each side = a cm. O is a point in the interior of, O ^, OE ^ and E F OF ^ such that O = cm, OE = 0 cm and OF = 6 cm. rea of equilateral triangle = a cm. lso area of = area of O + area of O + area of O a cm = = O + OE + (a + a 0 + a 6) cm OF a = 0a a = 60 ( a 0) a 0. \ rea of = (0 ) cm = 00 cm. Example 6. If the height of an equilateral triangle is 8 cm, calculate its area. Solution. Let be an equilateral triangle with side a cm. Let, then is midpoint of and = a cm. In, = 8 = a a ( height = = 8 cm given) 6 = a a a = 6 a = rea of = a cm = cm = 6 cm = 6 95 cm. Example 7. The base of an isosceles triangle is cm and its area is 9 sq. cm. Find its perimeter. Solution. Let be the isosceles triangle with base = cm and area = 9 sq. cm. Let h cm be its height and = = a cm. O Then area = base height 9 = h h = 6. In, = 99

5 (6) = a () ( = 56 = a a = 56 + a = 00 a = 0. Perimeter of = a + b = ( 0 + ) cm = 6 cm. = cm) Example 8. The base of an isosceles triangle measures cm and its area is 60 cm. Find its perimeter (using Heron s formula). Solution. Let each equal side of isosceles triangle be a cm, then s = a + a + cm = (a + ) cm. y Heron s formula, the area of triangle = ss ( a)( s b)( s c) ( a+ )( a+ a)( a+ a)( a+ ) = 60 (given) ( a+ ) ( a ) = 60 ( a+ )( a ) = 60 a = 5 a = 5 a = 69 a = ( a > 0) The perimeter of the triangle = cm + cm + cm = 50 cm. Example 9. Find the perimeter of an isosceles right-angled triangle whose area is 7 cm. Solution. Let be an isosceles right-angled triangle with = 90 and = = a cm. Then, area of = 7 = a a a = a cm a =. In, = 90. y Pythagoras theorem, we have a = + = a + a = a a cm a = a cm Perimeter of = + + = (a + a + a) cm = ( + ) a cm = ( + ) cm = ( + ) cm = 0 97 cm. Example 0. If the difference between the two sides of a right angled-triangle is cm and the area of the triangle is cm, find the perimeter of the triangle. Solution. Let be a right-angled triangle with = 90. Let = x cm, then = (x + ) cm. rea of = = x (x + ) x ( x + ) = 8 x + x 8 = 0 (x + 8) (x 6) = 0 x 00 Understanding ISE mathematics Ix

6 x = 8 or x = 6, but x cannot be negative x = 6. b = 6 cm, then = (6 + ) cm = 8 cm. In, = + a = (8) + (6) = = 00 a = 0 cm. Perimeter of = ( ) cm = cm. Example. The perimeter of a right-angled triangle is 60 cm. If its hypotenuse is 6 cm, find the area of the triangle. Solution. Let be a right-angled triangle with = 90, then its hypotenuse = 6 cm (given). Let base = x cm, then perimeter of = cm = + x cm + 6 cm = ( x) cm. In, = 90, ab + = ( x) + x = x + x + x = 676 x 68x + 80 = 0 x x + 0 = 0 (x ) (x 0) = 0 x =, 0. If x =, then = cm and = ( ) cm = 0 cm. area of = = 0 cm = 0 cm. If x = 0, then = 0 cm and = ( 0) cm = cm. area of = = 0 cm = 0 cm. Hence, the area of = 0 cm. Example. Each of equal sides of an isosceles triangle is cm greater than its height. If the base of the triangle is cm, find the area of triangle. Solution. Let be an isosceles triangle with base = cm. Let its height be x cm, then is mid-point of, therefore, = 6 cm. ccording to given, = = (x + ) cm. From right angle, by Pythagoras theorem, we get ab = + (x + ) = x + 6 x + x + = x + 6 x = x = 8. area of = = 8 cm = 8 cm. 0

Example. If the area of an isosceles triangle is 0 cm and the length of each of its equal sides is 7 cm, find its base. Solution. Let be an isosceles triangle with = = 7 cm and its area = 0 cm.

7 Example. If the area of an isosceles triangle is 0 cm and the length of each of its equal sides is 7 cm, find its base. Solution. Let be an isosceles triangle with = = 7 cm and its area = 0 cm. Let base = x cm. raw, then is mid-point of. bd = = x cm. In, = 90, ad + = ad = = 7 x ad = 89 x cm. rea of = 0 = x 89 x = x 89 x (0) = x (89 x ) x 89x + 00 = 0 (x 5) (x 6) = 0 x = 5, 6 x = 5, 8. ( x cannot be negative) base = = x cm = 0 cm or 6 cm. Example. In the adjoining figure, is an isosceles triangle with base = 8 cm and = = cm. is perpendicular to and O is a point on such that O = 90. Find the area of the shaded region. Solution. s, is mid-point of. bd = of 8 cm = cm. O From, by Pythagoras theorem, ad = = = 6 ad = 8 cm = 8 cm. area of = = 8 8 cm O O O = O. Let O = O = x cm. = cm. From O, by Pythagoras theorem, O + O = x + x = 8 x = 6 x =. (SS rule of congruency) 0 Understanding ISE mathematics Ix

8 area of O = O O = x x cm = x cm = cm = 6 cm. area of the shaded region = area of area of O = cm 6 cm = 6 ( ) cm = 9 5 cm. Exercise 5.. Find the area of a triangle whose base is 6 cm and corresponding height is cm.. Find the area of a triangle whose sides are : (i) cm, cm and 5 cm. (ii) 9 cm, 0 cm and cm. (iii) cm, 9 6 cm and 7 cm.. Find the area of a triangle whose sides are cm, 0 cm and cm. Hence, find the length of the altitude corresponding to the shortest side.. The sides of a triangular field are 975 m, 050 m and 5 m. If this field is sold at the rate of ` 000 per hectare, find its selling price. [ hectare = 0000 m ] 5. The base of a right angled triangle is cm and its hypotenuse is cm long. Find its area and the perimeter. 6. Find the area of an equilateral triangle whose side is 8 m. Give your answer correct to two decimal places. 7. If the area of an equilateral triangle is 8 cm, find its perimeter. 8. If the perimeter of an equilateral triangle is 6 cm, calculate its area and height. 9. (i) If the lengths of the sides of a triangle are in the ratio : : 5 and its perimeter is 8 cm, find its area. (ii) The sides of a triangular plot are in the ratio : 5 : 7 and its perimeter is 00 m. Find its area. Take = is a triangle in which = = cm and = 90. alculate the area of. also find the length of perpendicular from to. Hint: y Pythagoras theorem, = + = + = = cm.. Find the area of an isosceles triangle whose equal sides are cm each and the perimeter is 0 cm.. Find the area of an isosceles triangle whose base is 6 cm and perimeter is 6 cm.. The sides of a right-angled triangle containing the right angle are 5x cm and (x ) cm. alculate the length of the hypotenuse of the triangle if its area is 60 cm.. In, = 90, = (x + ) cm and = (x + ) cm. If the area of the is 60 cm, find its perimeter. 5. If the perimeter of a right angled triangle is 60 cm and its hypotenuse is 5 cm, find its area. 6. In, = 90 and is mid-point of. If = 0 cm and = 5 cm, find the area and the perimeter of. Hint: = = = = 9 cm. Use Pythagoras theorem to find. 0

Downloaded from

Downloaded from Exercise 12.1 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 perimeter is 180 cm,