Example 4
Kamla has a triangular field with sides 240 m, 200 m, 360 m, where she grew wheat. In another triangular field with sides 240 m, 320 m, 400 m adjacent to the previous field, she wanted to grow potatoes and onions. She divided the field in two parts by joining the mid-point of the longest side to the opposite vertex and grew potatoes in one part and onions in the other part. How much area
(in hectares) has been used for wheat, potatoes and onions? (1 hectare = 10000 m2)
E is the mid-point of AD
So, AE = ED = 200 m
Area of wheat field = Area ABC
Area of potato field = Area AEC
Area of onion field = Area EDC
Area ABC
Area of triangle = (s(s a)(s b)(s c))
Here, s is the semi-perimeter,
and a, b, c are the sides of the triangle
Let a = 200 m, b = 240 m, c = 360 m
s = ( + + )/2
Area ABC = (400(400 200)(400 240)(400 360)) m2
= (400 200 160 40) m2
= ((4 2 4 16) 106) m2
= ((4 4) 2 (16) 106) m2
= ((16) 2 (16) 106) m2
= ((16 16) 2 106) m2
= (162 2 106) m2
= 162 2 106
= 16000 2 m2
= (16000 2)/10000 hectares
= 1.6 x 2 hectares
= 1.6 x 1.414 hectares
= 2.26 hectares (approx.)
Area for growing wheat = 2.26 hectares
For finding Area AEC & Area CED
we need EC
But there is no possible way to find EC
Taking out ACD separately
Area of triangle = 1/2 Base Height
Triangles AEC & CED have the same base ( as AE = ED = 200m)
And they have the same height CF.
So, Area AEC = Area CED
Area AEC = Area CED = 1/2 Area ACD
Area ACD
Area of triangle = (s(s a)(s b)(s c))
Here, s is the semi-perimeter,
and a, b, c are the sides of the triangle
Let a = 240 m, b = 320 m, c = 400 m
s = ( + + )/2
Area of ACD = (480(480 240)(480 320)(480 400)) m2
= (480 240 160 80) m2
= ((48) (24) (16) (8) 104) m2
= ((4 12) (2 12) (16) (8) 104) m2
= ((12 12) (4) (16) (2 8) 104) m2
= ((12 12) (4) (16) (16) 104) m2
= ((122) (22) (162) 104) m2
= 122 22 162 104
= 12 2 16 102 m2
= 38400 m2
= 38400/10000 hectares
= 3.84 hectares
Thus, Area ACD = 3.84 hectares
So,
Area AEC = Area CED = 1/2 Area ACD
Area AEC = Area CED = 1/2 3.84 hectares
= 1.92 hectares
Hence,
Area for growing potatoes = 1.92 hectares
Area for growing onions= 1.92 hectares

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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