Question

The sides of a triangular plot are in ratio of 3:5:7 and its perimeter is 900m. Find it’s area

Answer 1

Given -

• Ratio of sides of a triangle = 3:5:7

• It's perimeter = 900 m

To find -

• Triangle's area

Formula used -

• Perimeter of Triangle

• Heron's formula

Solution -

In the question, we are provided with the ratio of sides of a triangular plot, and it's perimeter is 900m, and we need to find it's area. For that, first we will take the common ratio as x then, we will apply the formula of perimeter of Triangle, which will give us the value of sides of triangle, after that, we will apply Heron's formula, to find it's area. Let's do it!

Let -

The common ratio be x

So -

3 = 3x, side a

5 = 5x, side b

7 = 7x, side c

Perimeter of Triangle -

• P = a + b + c

On substituting the Values -

$\sf \: p \: = a \: + \: b \: + \: c \\ \\ \sf \: 900 \: m \: = 3x \: + \: 5x \: + \: 7x \\ \\ \sf \: 900 \: m \: = 15x \\ \\ \sf \: x \: = \dfrac{900}{15} \\ \\ \sf \: x \: = 60$

So -

$\sf \: 3x \: = \: 3 \: \times 60 \: = 180m \\ \\ \sf \: 5x \: = \: 5 \: \times \: 60 \: = 300m \\ \\ \sf \: 7x \: = \: 7 \: \times \: 60 \: = \: 420m$

Now -

First, we will find the semi-perimeter, and then, we will apply the heron's formula to find the area of the triangle.

Semi-perimeter -

$\sf \: s \: =\: \dfrac{perimeter}{2} \\ \\ \sf \: s\: = \: \dfrac{900}{2} \\ \\ \sf \: s \: = \: 450\:m$

Heron's formula -

$\sf \: \sqrt{s(s - a)(s - b)(s - c)}$

On substituting the values -

$\sf \: \sqrt{450(450 - 180)(450 - 300)(450 - 420)} \\ \\ \sf \: \sqrt{450(270)(150)(30)} \\ \\ \sf \: \sqrt{546750000} \\ \\ \sf \: 23382.68 { \: m}^{2}$

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Answer 2

${\large{\sf{\pmb{\underline{\maltese \: \: Undertanding \; the \; Question...}}}}}$

• This question says that we have to find the area of a triangular shaped plot whose perimeter is 900 metres. And it is also given that the sides of that triangular shaped plot are in the ratio as 3,5 and 7 means 3:5:7 and we also know that the traingle have 3 sides.

${\large{\sf{\pmb{\underline{\maltese \: \: Given \; that...}}}}}$

• The sides of a triangular plot are in ratio of 3:5:7

• Triangular plot perimeter is 900 metres.

${\large{\sf{\pmb{\underline{\maltese \: \: To \; find...}}}}}$

• Area of the triangular plot.

${\large{\sf{\pmb{\underline{\maltese \: \: Solution...}}}}}$

• Area of the triangular plot = 23382.68 m²

${\large{\sf{\pmb{\underline{\maltese \: \: Using \; concepts...}}}}}$

• Perimeter of traingle formula.
• Heron's formula.
• Semi perimeter formula

${\large{\sf{\pmb{\underline{\maltese \: \: Using \; formulas...}}}}}$

$\qquad \qquad \qquad {\pmb{\sf{\circ \: \: Perimeter \: of \: triangle \: formula}}}$

${\small{\underline{\boxed{\sf{a+b+c}}}}}$

$\qquad \qquad \qquad {\pmb{\sf{\circ \: \: Heron's \: formula}}}$

${\small{\underline{\boxed{\sf{\sqrt{s(s - a)(s - b)(s - c)}}}}}}$

$\qquad \qquad \qquad {\pmb{\sf{\circ \: \: Semi \: perimeter \: formula}}}$

${\small{\underline{\boxed{\sf{Semi \: perimeter \: = \dfrac{Perimeter}{2}}}}}}$

${\large{\sf{\pmb{\underline{\maltese \: \: Where,}}}}}$

• a denotes first side of triangle
• b denotes second side
• c denotes third side
• s denotes semi perimeter

${\large{\sf{\pmb{\underline{\maltese \: \: Assumptions...}}}}}$

~ As it's given that the sides of that triangular shaped plot are in the ratio as 3,5 and 7 means 3:5:7 Henceforth, we have to assume all the side of the triangle. So, assumptions are as follow:

• Common ratio as a
• 1st side as 3a
• 2nd side as 5a
• 3rd side as 7a

${\large{\sf{\pmb{\underline{\maltese \: \: Full \; Solution...}}}}}$

~ Firstly let us find the value of common ratio that is a by using the formula to find perimeter of triangle. We just have to put the values by using assumptions. Let's do it!

${\small{\underline{\boxed{\sf{P \: = a+b+c}}}}}$

${\sf{:\implies Perimeter \: = a+b+c}}$

${\sf{:\implies 900 = 3a + 5a + 7a}}$

${\sf{:\implies 900 = 8a + 7a}}$

${\sf{:\implies 900 = 15a}}$

${\sf{:\implies 900/15 = a}}$

${\sf{:\implies 60 = a}}$

${\sf{:\implies a = 60 \: m}}$

${\sf{:\implies Common \: ratio \: = 60 \: m}}$

${\underline{\frak{Henceforth, \: 60 \: m \: is \: the \: value \: of \: a \: (common \: ratio)}}}$

~ Now let's find the value of 1st side, 2nd side and 3rd side we just have to put 60 in the place of a. Let's do it!

$\qquad \qquad \qquad {\pmb{\sf{\bigstar \: \: Finding \: 1st \: side}}}$

${\sf{:\implies a \: = 60}}$

${\sf{:\implies 3a \: = 3(60)}}$

${\sf{:\implies 3 \times 60}}$

${\sf{:\implies 180 \: m}}$

${\underline{\frak{180 \: m \: is \: 1st \: side}}}$

$\qquad \qquad \qquad {\pmb{\sf{\bigstar \: \: Finding \: 2nd \: side}}}$

${\sf{:\implies a \: = 60}}$

${\sf{:\implies 5a \: = 5(60)}}$

${\sf{:\implies 5 \times 60}}$

${\sf{:\implies 300 \: m}}$

${\underline{\frak{300 \: m \: is \: 2nd \: side}}}$

$\qquad \qquad \qquad {\pmb{\sf{\bigstar \: \: Finding \: 3rd \: side}}}$

${\sf{:\implies a \: = 60}}$

${\sf{:\implies 7a \: = 7(60)}}$

${\sf{:\implies 7 \times 60}}$

${\sf{:\implies 420 \: m}}$

${\underline{\frak{420 \: m \: is \: 3rd \: side}}}$

~ Now we have to use heron's formula and have to put the values but firstly we have to find the value of s there. To find s we have to follow the formula!

${\small{\underline{\boxed{\sf{Semi \: perimeter \: = \dfrac{Perimeter}{2}}}}}}$

${\sf{:\implies Semi \: perimeter \: = \dfrac{Perimeter}{2}}}$

${\sf{:\implies Semi \: perimeter \: = \dfrac{900}{2}}}$

${\sf{:\implies Semi \: perimeter \: = 450 \: m}}$

${\underline{\frak{Henceforth, \: 450 \: m \: is \: semi \: perimeter}}}$

~ Now by using heron's formula let's put the values!

${\small{\underline{\boxed{\sf{\sqrt{s(s - a)(s - b)(s - c)}}}}}}$

${\sf{:\implies \sqrt{s(s - a)(s - b)(s - c)}}}$

${\sf{:\implies \sqrt{450(450 - 180)(450 - 300)(450 - 420)}}}$

${\sf{:\implies \sqrt{450(270)(150)(30)}}}$

${\sf{:\implies \sqrt{450 \times 270 \times 150 \times 30}}}$

${\sf{:\implies \sqrt{450 \times 270 \times 4500}}}$

${\sf{:\implies \sqrt{546750000}}}$

${\sf{:\implies 23382.68 \: m^{2}}}$

${\underline{\frak{Henceforth, \:23382.68 \: m^{2} \: is \: area}}}$