Question

The sides of a triangular plot are in the ratio of 3:5:7 and it's perimeter is 300m. Find its area.​

Answer 1

Given :–

• Sides of a triangular plot are in the ratio of 3:5:7
• It's perimeter = 300m

To Find :–

• Area of a triangular plot.

Solution :–

Let the sides of a triangular plot are 3x, 5x and 7x.

Perimeter = 300m

⟹ a + b + c = 300m

⟹ 3x + 5x + 7x = 300m

⟹ 8x + 7x = 300m

⟹ 15x = 300m

⟹ x = $\dfrac{300}{15}$

Cut the denominator (5) and the numerator (300) by 5, we obtain

⟹ x = $\dfrac{60}{3}$

Now, cut the denominator (3) and the numerator (60) by 3, we obtain

⟹ x = 20

So, the sides are :–

• 3x = 3 × 20 = 60m
• 5x = 5 × 20 = 110m
• 7x = 7 × 20 = 140m

Area of triangle = $\sqrt{s(s - a)(s-b)(s-c)}$

• a = 60
• b = 110
• c = 140

where, s = $\dfrac{a+b+c}{2}$

⟹ $\dfrac{60 + 100 + 140}{2}$

⟹ $\dfrac{300}{2}$

Cut the denominator (2) and the numerator (300) by 2, we obtain

⟹ 150

• s = 150

Now,

⟹ $\sqrt{150(150 - 60)(150 - 110)(150 - 140)}$

⟹ $\sqrt{150(90)(50)(10)}$

⟹ $\sqrt{(3 \times 5 \times 10)(3 \times 3 \times 10)(5 \times 10) \times 10}$

⟹ 3 × 5 × 10 × 10√3

⟹ 15 × 100√3

⟹ 1500√3m²

Hence,

The area of a triangular plot is 1500√3m².

Answer 2

$\large{\red{\underline{\tt{Digram\::-}}}}$

$\setlength{\unitlength}{1.5cm}\begin{picture}(6,8)\linethickness{0.5mm}\qbezier(1,.5)(2,1)(4,2)\qbezier(4,2)(2,3)(2,3)\qbezier(2,3)(2,3)(1,0.5)\put(.7, .3){ C }\put(4.05, 1.9){ B }\put(1.7, 2.95){ A }\put(3.2, 2.5){\sf{60 m}}\put(0.7,1.7){\sf{100 m}}\put(2.7, 1.05){\sf{140 m}}\end{picture}$

$\large{\red{\underline{\tt{Solution\::-}}}}$

Let the required ratio number be 3x, 5x and 7x

$\: :\implies\sf \:3x + 5x + 7x = Perimeter\\\\\\\::\implies\sf \:3x + 5x + 7x = 300\\\\\\:\implies\sf 15x = 300\\\\\\:\implies\sf x = \dfrac{300}{15}\\\\\\:\implies\sf x = 20\:m$

So,

• The first side, a = 3 × 20 = 60 m

• The second side, b = 5 × 20 = 100 m

• The third side, c = 7 × 20 = 140 m

⠀

☢ $\underline{\textsf{Semi - Perimeter of the Triangle :}}$

$:\implies\sf Semi \:Perimeter=\dfrac{Sum\:of\:Sides}{2}\\\\\\:\implies\sf s = \dfrac{a + b + c}{2}\\\\\\:\implies\sf s = \dfrac{60 + 100 + 140}{2}\\\\\\:\implies\sf s = \dfrac{300}{2}\\\\\\:\implies\sf s = 150\:m$

$\rule{120}{0.8}$

☢$\underline{\textsf{Area of a triangle :}}$

$\dashrightarrow\sf\:\:Area=\sqrt{s(s-a)(s-b)(s-c)}\\\\\\\dashrightarrow\sf\:\:Area=\sqrt{150(150-60)(150-100)(150-140)}\\\\\\\dashrightarrow\sf\:\:Area = \sqrt{150 \times 90 \times 50 \times 10}\\\\\\\dashrightarrow\:\:\underline{\boxed{\sf{Area = 1500\sqrt {3}\:m^2}}}$

⠀

$\therefore\:\underline{\bf{Volume\: of\: cylinder\: is \:1500\sqrt 3\:m^2}}.$