Question

Three horses are tethered with 7 m long ropes at the three corners of a triangular field having sides 20 m, 34m, and 42 m. Find the area of the plot.
i. Grazed by horses
ii. Remains ungrazed by horses

Answer 1

AnswEr :

• Sides of Field = 20m, 34m, 42m
• Three Horses with rope of 7m tethered at three corners of Triangular Field.

We will use Heron's Formula to find the Area of Triangle (let's say ∆ ABC)

⋆ Refrence of Image is in the Diagram :

$\setlength{\unitlength}{1cm}\begin{picture}(6,8)\linethickness{0.075mm}\put(1, .5){\line(2, 1){3}}\put(4, 2){\line(-2, 1){2}}\put(2, 3){\line(-2, -5){1}}\put(.7, .3){ A }\put(4.05, 1.9){ B }\put(1.7, 2.95){ C }\put(3.2, 2.5){ 20 m }\put(0.6,1.7){ 34 m }\put(2.7, 1.05){ 42 m }\end{picture}$

• First we will find the Semi Perimeter :

$\longrightarrow \tt Semi \:Perimeter = \dfrac{Sum \:of \:Sides}{2} \\ \\\longrightarrow \tt s = \dfrac{a + b + c}{2} \\ \\\longrightarrow \tt s = \dfrac{20 + 34 + 42}{2}\\ \\\longrightarrow \tt s = \cancel\dfrac{96}{2} \\ \\\longrightarrow \blue{\tt s = 48}$

• Calculation of Area of Triangle :

$\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{s(s - a)(s - b)(s - c)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{48(48 - 20)(48 - 34)(48- 42)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{48 \times28 \times14\times6}\\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{(6 \times 2\times4) \times(7 \times4) \times(7 \times2) \times6} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= \sqrt{(6 \times 6)(7 \times 7)(4 \times 4)(2 \times2)} \\ \\\longrightarrow \tt Area_{\tiny \triangle ABC}= 6\times7 \times 4 \times 2 \\ \\\longrightarrow \boxed{\orange{\tt Area_{\tiny \triangle ABC}= 336 \:{m}^{2}}}$

⠀

∴ Area of Triangular Field will be 336 m².

$\rule{300}{1}$

I ) Area of Plot Grazed by Horses.

Now, if these animals are tied at the corners will make Sector i.e. (3 Sectors of Radius 7 m), So will Find the Area that Animals can actually Graze.

• Area of Plot Grazed by Horses are :

$\longrightarrow \tt Area = \dfrac{\angle A}{360\degree}\pi {r}^{2} + \dfrac{\angle B}{360\degree}\pi {r}^{2} + \dfrac{\angle C}{360\degree}\pi {r}^{2} \\ \\\longrightarrow \tt Area = \pi {r}^{2} \bigg(\dfrac{\angle A}{360\degree}+ \dfrac{\angle B}{360\degree} + \dfrac{\angle C}{360\degree} \bigg)\\ \\\longrightarrow \tt Area = \pi {r}^{2} \bigg(\dfrac{\angle A +\angle B +\angle C}{360\degree}\bigg) \\ \\\longrightarrow \tt Area = \pi {r}^{2} \bigg( \cancel\dfrac{180 \degree}{360\degree}\bigg) \\ \\\longrightarrow \tt Area = \dfrac{ \cancel{22}}{\cancel7} \times \cancel7\times 7\times \dfrac{1}{\cancel2}\\\\\longrightarrow \tt Area = 11 \times 7\\ \\ \longrightarrow \blue{\tt Area =77 \: {m}^{2}}$

⠀

∴ Horses can Graze 77 m² Area of Plot.

$\rule{300}{2}$

II ) Remains ungrazed by Horses.

↠ Ungrazed Area = Total – Grazed Area

↠ Ungrazed Area = 336m² – 77m²

↠ Ungrazed Area = 259 m²

⠀

∴ Ungrazed Area of Plot will be 259 m².

#answerwithquality #BAL

Answer 2

$\huge{\green{\underline{\red{\mathscr{ANSWER:-}}}}}$

The area that can be grazed by the horse at each vertex is the area of the sector of radius 7 m at each vertex. 

To find that area we need to know the angle at each vertex.

We use the cosine rule in a triangle as we know the lengths of the sides.

AC² = AB² + BC² - 2 AB * BC * Cos B

20² = 34² + 42² - 2 * 34 * 42 * Cos B

Cos B = 0.88235   

  => B = 28.07⁰

AB² = AC² + BC² - 2 AC * BC * Cos C

34² = 42² + 20² - 2 * 42 * 20 * Cos C

Cos C = 0.6          => C = 53.13°

A = 180° - B - C = 98.80°

Area grazed by the horse at the vertex A = (π * 7²) * (98.80°/360°) m²

           = 42.247 m²

Area grazed by the horse at the vertex B =  (π * 7² * (28.07°/360°) m²

         = 12 m²

Area grazed by the horse at the vertex C = (π 7² * (53.13°/360°) m²

       = 22.718 m²

Total area of the triangle ABC can be found by Heron's formula as:

  s = semi perimeter = (AB+BC+CA)/2 =  48 m

area of ΔABC,

$\sqrt{s(s - a)(s - b)(s - c)} \\ \sqrt{48 \times 6 \times 28 \times 14} \\ = 336cm {}^{2}$

The area left ungrazed is = 336 - 22.718 - 12 - 42.247 = 259.035 m².

$\huge{\purple{\underline{\pink{\mathscr{THANKS.}}}}}$