Question

The perimeter of a triangular field is 540 m. and its sides are in ratio 25:17:12. Find the area of the triangle?

_R.D. Sharma

Answer 1

Answer:

Question :-

• The perimeter of a triangular field is 540m and its sides are in the ratio 25:17:12.Find the area of the triangle.

Answer :-

• Area of triangle is 9000 m^2.

Given :-

• The perimeter of triangular field is 540m and its sides are in ratio 25:17:12.

To find :-

• Find the Area of triangle

Solution :-

• Here according to the given question we can verify that,

• Perimeter =25x+17x+12x
• 540 =54x
• $x = \frac{540}{54}$
• x=10m.

Now ,

• The sides of triangle
• AB=25x=25×10=250m
• BC=17x=17×10=170 m.
• AC=12x=12×10=120m.

Lets ,

• Take first Semi perimeter of triangular field =
• $\frac{540}{2} = 270m$

By using Heron's formula we get the answer for your question.

• The Heron's formula is,

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

• Now applying all the values we get,

• $\sqrt{270(270 - 250)(270 - 170)(270 - 120)}$

• $\sqrt{270 \times 20 \times 100 \times 150}$
• $\sqrt{ {3}^{4} \times {10}^{5} \times 2 \times 5} {m}^{2}$

• $= \sqrt{( {3}^{2} \times {10}^{(3 {)}^{2} } } {m}^{2}$

• $= 9 \times 1000 {m}^{2}$

• $area \: of \: triangle = 9000 {m}^{2}$

Used formulae:-

• Perimeter of triangle
• Herons formula
• we used to get the answer.

Hope it helps u mate .

Thank you .

Answer 2

★ Assumption Needed:-    

Let the sides be,

• ➾  Side a = 25x m
• ➾  Side b = 17x m
• ➾  Side c = 12 m

★ To Calculate:-    

• Area of the triangle.

★ Formula Used:-    

~Perimeter of the triangle

• $\Large\boxed{\sf{Perimeter_{(\:TRIANGLE\:)}\:=\:a\:+\:b\:+\:c}}$

Where,

• a,b and c are sides of the triangle

~Semi-Perimeter

• $\Large\boxed{\sf{Semi\:-\:Perimeter\:=\:\left(\:\dfrac{a\:+\:b\:+\:c}{2}\:\right)}}$

Where,

• ➾  a = Length of side a
• ➾  b = Length of side b
• ➾  c = Length of side c

~Area of the triangle

• $\Large\boxed{\underline{\sf{Area\:_{(\:TRIANGLE\:)}\:=\:\sqrt{s\:(\:s\:-\:a\:)\:(\:s\:-\:b\:)\:(\:s\:-\:c\:)}} }}$

Where,

• ➾  s = semi-perimeter
• ➾  a = 8 cm = Length of side a
• ➾  b = 42 cm = Length of side b
• ➾  c = 44 cm = Length of side c

★ Calculating:-      

Step1: First we will find out the sides of the triangle by substituting the values in the formula [ Perimeter of the triangle = a + b + c ]

$\qquad\sf{:\implies\:Perimeter_{(\:TRIANGLE\:)}\:=\:a\:+\:b\:+\:c}$

$\qquad\sf{:\implies\:540\:m\:=\:25x\:+\:17x\:+\:12x}$

$\qquad\sf{:\implies\:540\:m\:=\:54\:x}$

$\qquad\sf{:\implies\:x\:=\:\dfrac{540}{54}}$

$\qquad\sf{:\implies\:x\:=\:10\:}$

~So the sides are,

• Side a = 25x = 25 × 10 = 250 m
• Side b = 17x = 17 × 10 = 170 m
• Sides c = 12x = 12 × 10 = 120m

$. \:°\: .\underline{\sf{The\: sides\: are\: 250\: m\:,\: 170 \:m\:,\: 120\: m\:.}}$

Step2: Now we will find out the semi-perimeter so we will substitute the values in the formula. [ Semi-perimeter = a + b + c/2 ]

$\qquad\sf{:\implies\:Semi\:-\:Perimeter\:=\:\left(\:\dfrac{a\:+\:b\:+\:c}{2}\:\right)}$

$\qquad\sf{:\implies\:Semi\:-\:Perimeter\:=\:\left(\:\dfrac{250\:+\:170\:+\:120}{2}\:\right)}$

$\qquad\sf{:\implies\:Semi\:-\:Perimeter\:=\:\left(\:\dfrac{540}{2}\:\right)}$

$\qquad\sf{:\implies\:Semi\:-\:Perimeter\:=\:270\:m}$

$. \:°\: .\underline{\sf{The\: semi\:-\:perimeter\:of\:the\: triangle\:is\: 270\:m\:.}}$

Step3: Now to calculate the triangle area, we will again use the formula and substitute the values. [ Area of triangle = √s ( s - a ) ( s -b ) ( s - c )]

$\qquad\sf{:\implies\:Area\:_{(\:\:TRIANGLE)}\:=\:\sqrt{s\:(\:s\:-\:a\:)\:(\:s\:-\:b\:)\:(\:s\:-\:c\:)} }$

$\qquad\sf{:\implies\:Area\:_{(\:\:TRIANGLE)}\:=\:\sqrt{270\:(\:270\:-\:250\:)\:(\:270\:-\:170\:)\:(\:270\:-\:120\:)} }$

$\qquad\sf{:\implies\:Area\:_{(\:\:TRIANGLE)}\:=\:\sqrt{270\:(\:20\:)\:(\:100\:)\:(\:150\:)} }$

$\qquad\sf{:\implies\:Area\:_{(\:\:TRIANGLE)}\:=\:\sqrt{270\:\times\:20\:\times\:100\:\times\:150\:} }$

$\qquad\sf{:\implies\:Area\:_{(\:\:TRIANGLE)}\:=\:\sqrt{8,10,00,000} }$

$\qquad\sf{:\implies\:Area\:_{(\:\:TRIANGLE)}\:=\:9000\:m}$

$. \:°\: .\underline{\sf{The\: area\:of\:the\: triangle\:is\: 9000\:m^{2}\:.}}$

★ Answer:-      

• . ° . The area of the triangle is 9000 m².