Question

14.The sides of a triangular plot are in the ratio 2:3:4 and its perimeter is 450m find
its area.

Answer 1

Given: Ratio of sides of a triangular plot is 2:3:4 & it's perimeter is 450 m.

To find: Area of triangular plot?

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀⠀

☯ Let sides of triangular plot be 2x, 3x and 4x.

⠀⠀

$\underline{\bigstar\:\boldsymbol{Using\: Heron's\:formula\::}}$

$\star\;{\boxed{\sf{\pink{Area_{\;(triangle)} = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

$\sf Here \begin{cases} & \sf{a = \bf{2x}} \\ & \sf{b = \bf{3x}} \\ & \sf{c = \bf{4x}} \end{cases}$

$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

$:\implies\sf s = \dfrac{Perimeter}{2}$

$:\implies\sf s = \cancel{\dfrac{450}{2}}$

$:\implies{\underline{\boxed{\frak{\purple{s = 225\:m}}}}}\;\bigstar$

Therefore,

⠀⠀⠀⠀

$:\implies\sf 225 = \dfrac{2x + 3x + 4x}{2}$

$:\implies\sf 225 = \dfrac{9x}{2}$

$:\implies\sf 9x = 450$

$:\implies\sf x = \cancel{ \dfrac{450}{9}}$

$:\implies{\underline{\boxed{\frak{\purple{x = 50}}}}}\;\bigstar$

$\dag\;{\underline{\frak{Sides\:of\:triangle\:are,}}}$

• a = 2x = 2 × 50 = 100 m
• b = 3x = 3 × 50 = 150 m
• c = 4x = 4 × 50 = 200 m

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$\therefore\:{\underline{\sf{Sides\;of\;triangle\:area\;100\;m,\:150\;m\:and\;200\;m}}}.$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀⠀

$\dag\;{\underline{\frak{Now,\:Putting\:values\:in\;formula,}}}$

$:\implies\sf \sqrt{225(225 - 100)(225 - 150)(225 - 200)}$

$:\implies\sf \sqrt{225 \times 125 \times 75 \times 25}$

$:\implies\sf \sqrt{(15 \times 15) \times (25 \times 5) \times (25 \times 3) \times (5 \times 5)}$

$:\implies\sf \sqrt{15^2 \times 5^2 \times 5 \times 5^2 \times 3 \times 5^2}$

$:\implies\sf 15 \times 5 \times 5 \times 5 \times \sqrt{5 \times 3}$

$:\implies\sf 1875 \times \sqrt{15}$

$:\implies{\underline{\boxed{\frak{\pink{ 1875 \sqrt{15} \:m^2}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Area\:of\:triangular\:plot\:is\; \bf{1875 \sqrt{15} \:or\:7261.843\:m^2}.}}}$

Answer 2

Answer:

Given :-

• Sides of triangular plot = 2:3:4
• Perimeter = 450 m

To Find :-

• Area

Solution :-

Let sides be x

Firstly let's find all sides of triangle

As we know that Perimeter is the sum of all sides

$\sf \: 2x + 3x + 4x = 450$

$\sf \: 9x = 450$

$\sf \: x = \dfrac{450}{9}$

$\sf \: x = 50$

Let's find angles

$\sf \: 2(50) = 100$

$\sf3(50) = 150$

$\sf \: 4(50) = 200$

Now,

Let's find its semiperimeter

$\sf \: s = \dfrac{a + b + c}{2}$

$\sf \: s = \dfrac{100 + 150 + 200}{2}$

$\sf \: s = \dfrac{450}{2} = 225$

Now,

Let's find Area by herons formula.

$\huge \bf \green{\sqrt{s(s - a)(s - b)(s - c)} }$

$\tt \mapsto \sqrt{225(225- 100)(225 - 150)(225 - 200)}$

$\tt \mapsto \: \sqrt{225\times 125\times 75 \times 25}$

$\tt \mapsto \: \sqrt{(15 \times 15) (25 \times 5)(25 \times 3)(5 \times 5)}$

$\tt\sqrt{15 {}^{2} \times 5{}^{2} \times 5 \times {5}^{2} \times }$

$\tt \: 1875× \sqrt{15}$

$\huge \tt \mapsto 1875× \sqrt{15}{m}^{2}$

Diagram :-

$\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\thicklines\qbezier(1, 0)(1,0)(3,3)\qbezier(5,0)(5,0)(3,3)\qbezier(5,0)(1,0)(1,0)\put(2.85,3.2){ \bf A }\put(0.5,-0.3){ \bf C }\put(5.2,-0.3){ \bf B }\end{picture}$