Question

Apply Heroe's formula and solve the following question.


Questions.

(i) a=30m b=24m c=20m
(ii)a=40m b=42m c=35m

Hero's Formula : √S(S-A) (S-B) (S-C)

S= Semiperimeter
​

Answer 1

AnswEr :

( i ) a = 30m, b = 24m, c = 20m

⋆ Refrence of Image is in the Diagram :

$\setlength{\unitlength}{1cm}\begin{picture}(6,8)\linethickness{0.075mm}\put(1, .5){\line(2, 1){3}}\put(4, 2){\line(-2, 1){2}}\put(2, 3){\line(-2, -5){1}}\put(.7, .3){ A }\put(4.05, 1.9){ B }\put(1.7, 2.95){ C }\put(3.2, 2.5){ 20 m }\put(0.6,1.7){ 24 m }\put(2.7, 1.05){ 30 m }\end{picture}$

• First we will find the Semi Perimeter :

$\begin{lgathered}\longrightarrow \tt Semi \:Perimeter = \dfrac{Sum \:of \:Sides}{2} \\ \\\longrightarrow \tt s = \dfrac{a + b + c}{2} \\ \\\longrightarrow \tt s = \dfrac{30 + 24 + 20}{2}\\ \\\longrightarrow \tt s = \cancel\dfrac{74}{2} \\ \\\longrightarrow \blue{\tt s = 37}\end{lgathered}$

• Calculation of Area of Triangle :

$\begin{lgathered}\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{s(s - a)(s - b)(s - c)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{37(37 - 30)(37 - 24)(37- 20)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{37 \times7 \times13\times17}\\ \\ \longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{57239} \\ \\\longrightarrow \boxed{\red{\tt Area_{\tiny \triangle ABC} \approx 239.25\:{m}^{2}}}\end{lgathered}$

⠀

∴ Hence, Area of Triangle is 239.25 m²

$\rule{300}{2}$

( ii ) a = 40m, b = 42m, c = 35m

⋆ Refrence of Image is in the Diagram :

$\setlength{\unitlength}{1cm}\begin{picture}(6,8)\linethickness{0.075mm}\put(1, .5){\line(2, 1){3}}\put(4, 2){\line(-2, 1){2}}\put(2, 3){\line(-2, -5){1}}\put(.7, .3){ A }\put(4.05, 1.9){ B }\put(1.7, 2.95){ C }\put(3.2, 2.5){ 35 m }\put(0.6,1.7){ 40 m }\put(2.7, 1.05){ 42 m }\end{picture}$

• First we will find the Semi Perimeter :

$\begin{lgathered}\longrightarrow \tt Semi \:Perimeter = \dfrac{Sum \:of \:Sides}{2} \\ \\\longrightarrow \tt s = \dfrac{a + b + c}{2} \\ \\\longrightarrow \tt s = \dfrac{40 + 42 + 35}{2}\\ \\\longrightarrow \tt s = \cancel\dfrac{117}{2} \\ \\\longrightarrow \blue{\tt s = 58.5}\end{lgathered}$

• Calculation of Area of Triangle :

$\begin{lgathered}\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{s(s - a)(s - b)(s - c)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{58.5(58.5 - 40)(58.5 - 42)(58.5- 35)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{58.5 \times18.5 \times16.5\times23.5}\\ \\ \longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{419642.438} \\ \\\longrightarrow \boxed{\red{\tt Area_{\tiny \triangle ABC} \approx 647.8\:{m}^{2}}}\end{lgathered}$

⠀

∴ Hence, Area of Triangle is 647.8 m²

◗ NOTE : These Values are Approx.

#answerwithquality #BAL

Answer 2

$\bf{\Huge{\underline{\boxed{\bf{\blue{ANSWER\::}}}}}}$

$\bf{\Large{\underline{\bf{\orange{First\:Case\::}}}}}$

$\bf{Given,\:side\:of\:triangle\:are\begin{cases}\sf{A=30m}\\ \sf{B=24m}\\ \sf{C=20m}\end{cases}}$

• $\bf{\huge{\underline{\sf{\pink{Using\:Heron's\:formula\::}}}}}$

→ Semi-perimeter = $\bf{\frac{A+B+C}{2} }$

→ Semi-perimeter = $\bf{\frac{30m+24m+20m}{2} }$

→ Semi-perimeter = $\bf{\cancel{\frac{74m}{2}}}$

→ Semi-perimeter,[S] = 37m

• $\bf{\large{\underline{\sf{\green{Area\:of\:ΔABC}}}}}$

→ Area of Δ = $\bf{\sqrt{S(S-A)(S-B)(S-C)} }$

→ Area of Δ = $\bf{\sqrt{37(37-30)(37-24)(37-20)} }$

→ Area of Δ = $\bf{\sqrt{37(7)(13 )(17)}}$

→ Area of Δ = $\bf{\sqrt{37*7*13*17} }$

→ Area of Δ = $\bf{\sqrt{57239} }$

→ Area of Δ = 239.24m²

Thus,

The area of triangle is 239.24m².

$\bf{\Large{\underline{\bf{\orange{Second\:Case\::}}}}}$

$\bf{Given,\:side\:of\:triangle\:are\begin{cases}\sf{A=40m}\\ \sf{B=42m}\\ \sf{C=35m}\end{cases}}$

• $\bf{\huge{\underline{\sf{\pink{Using\:Heron's\:formula\::}}}}}$

→ Semi-perimeter = $\bf{\frac{A+B+C}{2} }$

→ Semi-perimeter = $\bf{\frac{40m+42m+35m}{2} }$

→ Semi-perimeter = $\bf{\cancel{\frac{117m}{2} }}$

→ Semi-perimeter,[S] = 58.5m

Now,

⇒ Area of Δ = $\bf{\sqrt{S(S-A)(S-B)(S-C)} }$

⇒ Area of Δ = $\bf{\sqrt{58.5(58.5-40)(58.5-42)(58.5-35)} }$

⇒ Area of Δ = $\bf{\sqrt{58.5(18.5)(16.5)(23.5)} }$

⇒ Area of Δ = $\bf{\sqrt{419642.4375} }$

⇒ Area of Δ = 647.80m²

Thus,

The area of triangle is 647.80m².