Question

Two triangular parks are situated side by side. A gardener while planting along the periphery of the park, found that the sides of the first triangular park are in ratio 3:5:7, and walked 90 m to complete planting along the periphery. For the second triangular park, he found the ratio of sides as 3:4:5 and walked 60 m to complete planting along the periphery. Calculate the difference in areas of the two parks.

• A. 73.72 $m^2$​
• B. 83.83 $m^2$​
• C. 93.92 $m^2$​
• D. 53.92 $m^2$​

Answer 1

The correct option is B

83.83 $m^2$​

In the first triangular park, gardener walks 90 m along the periphery, which means, the perimeter of the park is 90 m.
Given, the sides are in the ratio 3:5:7.
So, let the lengths of the sides be 3x, 5x and 7x.
Then, $3x+5x+7x=90$
$\Rightarrow 15x=90\Rightarrow x=6$
So, the lengths of sides of the park are 18 m, 30 m and 42 m.
Area (A) of the triangle can be calculated using Heron's formula, given by:
$A=\sqrt{s(s-a)(s-b)(s-c)}$
where, s is the semiperimeter ans a, b and c are the sides.
Semi-perimeter (s) of the first park
$=\frac{(18+30+42)}{2}$ = 45 m
So, the area ($A_1$) of the first triangular park is given by
$A_1=\sqrt{45(45-18)(45-30)(45-42)}=\sqrt{45\times 27\times 15\times 3}⟹A_1=233.83m^2$

Similarly, for the second triangular park, we have:
$3x+4x+5x=60⟹x=5$
So, the lengths of the sides would be 15 m , 20 m and 25 m.
Semi- perimeter = $\frac{(15+20+25)}{2}$ = 30 m
So, the area ($A_2$) of the second triangular park is given by
$A_2=\sqrt{30(30-15)(30-20)(30-25)}=\sqrt{30\times 15\times 10\times 5}⟹A_2=150m^2$

Thus, the difference between the areas is,
$A_1-A_2=233.83-150=83.83m^2$.