Question

The sides of a triangular plot are in the ratio of 6 : 7: 8 and its
perimeter is 420 m. Find its area.
​

Answer 1

Step-by-step explanation:

$\huge\mathrm\star\fbox{Answer:-}\star$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\huge\mathrm\star\fbox{Solution:-}\star$

$\Rightarrow$ $\mathrm{\boxed{The\ area\ of\ the\ triangular\ field\ is\ 2100\ \sqrt{15}m²}}$

$\huge\mathrm\star\fbox{Explanation:-}\star$

:$\Longrightarrow$ $\mathrm{Given,\ perimeter\ of\ a\ triangular\ field\ is\ 420m\ and\ its\ sides\ are\ in\ the\ ratio\ 6\ :\ 7\ :\ 8.}$

:$\Longrightarrow$ $\mathrm{Let\ sides\ of\ a\ triangular\ field\ be\ a\ =\ 6x,\ b\ =\ 7x\ and\ c\ =\ 8x.}$

:$\Longrightarrow$ $\mathrm{Perimeter\ of\ a\ triangular\ field,\ 2s\ =\ a\ +\ b\ +\ c}$

:$\Longrightarrow$ $\mathrm{420\ =\ 6x\ +\ 7x\ +\ 8x\ =\ 420\ =\ 21x}$

:$\Longrightarrow$ $\mathrm{x\ =\ \dfrac{420}{21}\ =\ 20m}$

:$\Longrightarrow$ $\therefore$ $\mathrm{Area\ of\ a\ triangular\ field\ are}$

:$\Longrightarrow$ $\mathrm{a\ =\ 6\ ×\ 20\ =\ 120m}$

:$\Longrightarrow$ $\mathrm{b\ =\ 7\ ×\ 20\ =\ 140m}$

:$\Longrightarrow$ $\mathrm{c\ =\ 8\ ×\ 20\ =\ 160m}$

:$\Longrightarrow$ $\mathrm{\dfrac{420}{2}\ =\ 210m}$

:$\Longrightarrow$ $\mathrm{Area\ of\ triangular\ field}$

:$\Longrightarrow$ $\mathrm{\sqrt{s(s-a)(s-b)(s-c)}}$ $\mathrm{by,\ herons\ formula}$

:$\Longrightarrow$ $\mathrm{\sqrt{210(210-120)(210-140)(210-160)}}$

:$\Longrightarrow$ $\mathrm{100\ \sqrt{21×9×7×5}}$

:$\Longrightarrow$ $\mathrm{100\ \sqrt{7×3×3²×7×5}}$

:$\Longrightarrow$ $\mathrm{100\ ×\ 7\ ×\ 3\ ×\ \sqrt{15}\ =\ 2100\ \sqrt{15m²}}$

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$\Large\mathrm\star\fbox{Hence,\ solved}\star$

Answer 2

$\huge {\bf {\underbrace {Question}}}$

The sides of a triangular plot are in the ratio of 6 : 7: 8 and its

perimeter is 420 m. Find its area.

$\huge {\bf {\underbrace {Answer}}}$

Given:-

• Sides of a triangular plot are in the ratio of 6:7:8

• Perimeter is 420 m.

To Find:-

• Area of a triangular plot.

Solution:-

Let, sides in the ratio 6:7:8 be a, b and c

Let,

a = 6x

b = 7x

c = 8x

Now,

${\sf {Perimeter\: of\: triangle = 420 m.}}$

${\sf {⟹}}$ ${\sf {a + b + c = 420}}$

${\sf {⟹}}$ ${\sf {6x + 7x + 8x = 420}}$

${\sf {⟹}}$ ${\sf {21x = 420}}$

$\large {\sf {⟹}}$ ${\sf {x = \frac{420}{21}}}$

${\sf {⟹}}$ ${\sf {\boxed {\green {x = 20m.}}}}$

So, the lines will be

a = 6x = 6 × 20 = 120 m.

b = 7x = 7 × 20 = 140 m.

c = 8x = 8 × 20 = 160 m.

Then,

${\sf {⟹}}$ ${\large {\sf {s = \frac{a + b + c}{2}}}}$

${\sf {⟹}}$ ${\large {\sf {s = \frac {120 + 140 + 160}{2}}}}$

${\sf {⟹}}$ ${\large {\sf {s = \frac {420}{2}}}}$

${\sf {⟹}}$ ${\sf {\boxed {\green {s = 210}}}}$

Area of triangle by Heron's formula

${\sf {\boxed {\green {Area = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

${\sf {⟹}}$ ${\sf {\sqrt{210(210 - 120)(210 - 140)(210 - 160)}}}$

${\sf {⟹}}$ ${\sf {\sqrt{210(90)(70)(50}}}$

${\sf {⟹}}$ ${\sf {\sqrt{66150000}}}$

${\sf {⟹}}$ ${\sf {\boxed {\green {{8133.27m}^{2}}}}}$

${\sf {\boxed {\pink {Hence,\: the\: area\: of\: triangle\: is\: {8133.27m}^{2}}}}}$