Suman has a piece of land, which is in the shape of rhombus. She divides the land in two
equal parts by drawing a diagonal. Its perimeter is 400m and one of the diagonals is of length
120m.
a) Find the side of the land.
b) Find the perimeter of triangle ABD.
c) What is the area of triangle CBD?
d) Find the area of whole land.
e) Find the value of semi perimeter S.
Answers
Given:
- Perimeter of rhombus park = 400m
- Length of one diagonal = 120m
- A rhombus ABCD with diagonals AC and BD where, BD =120m
To Find:
- Side of land
- Perimeter of ∆ ABD
- Area of ∆ CBD
- Area of whole land
- Value of semi perimeter S.
Solution:
➥ Finding side of the land-
Let the side of rhombus be x
According to given conditions;
Perimeter of rhombus = 400m
➠ 4 × Side = 400m
➠ Side = 400 ÷ 4
➠ Side = 100m
∴ Side = 100m
➥ Finding perimeter of ∆ ABD
In ∆ ABD
AB = Side = 100m
BD = Diagonal = 120m
DA = Side = 100m
Perimeter of ∆ ABD
= Sum of all sides
= AB + BD + DA
= 100 + 120 + 100
= 320
∴ Perimeter of ∆ ABD is 320m.
➥ Finding Area of ∆ CBD
In ∆ CBD
CB = 100m
CD = 100m
BD = 120m
Value of s = (Sum of sides) ÷ 2
= (100 + 100 + 120) ÷ 2
= 320 ÷ 2
= 160
Hence, Area of ∆ CBD by Heron's Formula
= √ s (s - a) (s - b) (s - c)
= √ 160 (160 - 100) (160 - 100) (160 - 120)
= √ 2 × 2 × 40 × 60 × 60 × 40
= 2 × 40 × 60
= 4800 m²
∴ Area of ∆ CBD is 4800 m²
➥ Finding Area of whole land
Area of whole land
= Area of ∆ CBD + Area of ∆ ABD
In ∆ CBD and ∆ ABD
CB = AB [Equal sides]
BD = BD [Common]
AD = CD [Equal sides]
∴ ∆ CBD ≅ ∆ ABD
So, Area of ∆ CBD and ∆ ABD are also equal
Hence,
Area of whole land
= Area of ∆ CBD + Area of ∆ ABD
= 4800 + 4800
= 9600 m²
∴ Area of whole land is 9600m²
➥ Finding Semi perimeter (Value of S)
Semiperimeter
= Perimeter ÷ 2
= 400 ÷ 2
= 200m
∴ Value of S is 200m.
Answer:-
- Side = 100m
- Perimeter of ∆ ABD is 320m.
- Area of ∆ CBD is 4800 m²
- Area of whole land is 9600m²
- Value of S is 200m.
