Question

do it with explanation​

Answer 1

GivEn:-

• The Perimeter of a triangle is 450 m.
• The ratio of sides of triangle are 13:12:15.

⠀⠀⠀⠀⠀⠀⠀

To find:-

• Area of triangle using Heron's formula.

⠀⠀⠀⠀⠀⠀⠀

Solution:-

⠀⠀⠀⠀⠀⠀⠀

☯ Let sides of triangle be 13x, 12x and 5x.

⠀⠀⠀⠀⠀⠀⠀

$\sf where \begin{cases} & \sf{a = \bf{13\;x}} \\ & \sf{b = \bf{12\;x}} \\ & \sf{c = \bf{5\;x}} \end{cases}$

$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

The Perimeter of a triangle is 450 m.

⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf 13x + 12x + 5x = 450$

$:\implies\sf 30x = 450$

$:\implies\sf x = \cancel{ \dfrac{450}{30}}$

$:\implies{\boxed{\frak{\pink{x = 15}}}}\;\bigstar$

⠀⠀━━━━━━━━━━━━━━━━━━━━━

Therefore,

⠀⠀⠀⠀⠀⠀⠀

• a = 13x = 195 m

• b = 12x = 180 m

• c = 5x = 75 m

⠀⠀⠀⠀⠀⠀⠀

${\underline{\sf{\bigstar\;Using\; Heron's\;Formula\;:}}}$

We know that,

⠀⠀⠀⠀⠀⠀⠀

$\rightarrow\sf s = \dfrac{a + b + c}{2}\qquad\qquad\bigg\lgroup\bf s = semi - perimeter \bigg\rgroup$

$\rightarrow\sf s = \dfrac{195 + 180 + 75}{2}$

$\rightarrow\sf s = \cancel{ \dfrac{450}{2}}$

$\rightarrow\bf \red{s = 225}$

Now,

⠀⠀⠀⠀⠀⠀⠀

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

$:\implies\sf \sqrt{225(225 - 195)(225 - 180)(225 - 75)}$

$:\implies\sf \sqrt{225(30)(45)(150)}$

$:\implies\sf \sqrt{225 \times 202500}$

$:\implies\sf \sqrt{45562500}$

$:\implies\sf{\boxed{\frak{\pink{6750\;m^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;Area\;of\;triangle\;is\; \bf{6750}.}}}$

Answer 2

GIVEN:-

The Perimeter of a triangle is 450 m.

The ratio of sides of triangle are 13:12:15.

⠀⠀⠀⠀⠀⠀⠀

To find:-

Area of triangle using Heron's formula.

⠀⠀⠀⠀⠀⠀⠀

Solution:-

⠀⠀⠀⠀⠀⠀⠀

☯ Let sides of triangle be 13x, 12x and 5x.

⠀⠀⠀⠀⠀⠀⠀

$\sf where \begin{cases} & \sf{a = \bf{13\;x}} \\ & \sf{b = \bf{12\;x}} \\ & \sf{c = \bf{5\;x}} \end{cases}$

$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

The Perimeter of a triangle is 450 m.

⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf 13x + 12x + 5x = 450$

$:\implies\sf 30x = 450$

$:\implies\sf x = \cancel{ \dfrac{450}{30}}$

$:\implies{\boxed{\frak{\pink{x = 15}}}}\;\bigstar$

⠀⠀━━━━━━━━━━━━━━━━━━━━━

Therefore,

⠀⠀⠀⠀⠀⠀⠀

a = 13x = 195 m

b = 12x = 180 m

c = 5x = 75 m

⠀⠀⠀⠀⠀⠀⠀

${\underline{\sf{\bigstar\;Using\; Heron's\;Formula\;:}}}$

We know that,

⠀⠀⠀⠀⠀⠀⠀

$\rightarrow\sf s = \dfrac{a + b + c}{2}\qquad\qquad\bigg\lgroup\bf s = semi - perimeter \bigg\rgroup$

$\rightarrow\sf s = \dfrac{195 + 180 + 75}{2}$

$\rightarrow\sf s = \cancel{ \dfrac{450}{2}}$

$\rightarrow\bf \red{s = 225}$

Now,

⠀⠀⠀⠀⠀⠀⠀

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

$:\implies\sf \sqrt{225(225 - 195)(225 - 180)(225 - 75)}$

$:\implies\sf \sqrt{225(30)(45)(150)}$

$:\implies\sf \sqrt{225 \times 202500}$

$:\implies\sf \sqrt{45562500}$

$:\implies\sf{\boxed{\frak{\pink{6750\;m^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;Area\;of\;triangle\;is\; \bf{6750}.}}}$