Question

three horses are tethered with 7 metre long ropes at the three corners of a triangular field having sides 20m,34m and 42m.find the area of the plot wch can be grazed by the horses.also find the area of the plot wch remains ungrazed.
please reply as soon as possible

Answer 1

Hope it is helpful......

Answer 2

Let

• $∠A=ፀ_1°,$
• $∠B = ፀ_2°\: and$
• $∠C= ፀ_3°$

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Area which can be grazed by the horses= sum of the the areas of three sectors with Central angles $\sf ፀ_1°,ፀ_2°and \:ፀ_3°$, and each with radius 7 m

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$\:\:\:\:\:\:\:\:$$\mapsto$$\bigg($ $\bf\dfrac{πr²ፀ_1}{360}$+$\bf\dfrac{πr²ፀ_2}{360}$+$\bf\dfrac{πr²ፀ_3}{360}$$\bigg)$$\sf m²$

$\sf \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:|where,r=7\:m|$

$\:$

$\:\:\:\:\:\:\:\:\:\:\:\:\:$$\mapsto$$\bf\dfrac{πr²}{360}$$\sf (ፀ_1+ፀ_2+ፀ_3)m²$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\mapsto$$\bigg($$\bf\dfrac{πr²×180}{360}$$\bigg)$$\sf m²$

$\sf \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:[\becauseፀ_1+ፀ_2+ፀ_3=∠A+∠B+∠C=180°]$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:$$\mapsto$$\bigg($ $\bf\dfrac{22}{7}$ $\sf ×7×7×$ $\bf\dfrac{1}{2}$ $\bigg)$ $\sf m²$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\underline{\boxed{\sf{=\frak{\red{77m².}}}}}$

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$\textbf{Sides\:of\:the\:plot\:are-}$

$\:\:\:\:\:\:\:\:$$\large\mathtt{a=20m,\:b=34m,\:c=42m.}$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:$$\therefore\:$ $\sf s\:=\:$ $\bf\dfrac{a+b+c}{2}$ $=48m$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\leadsto$$\sf (s-a)=28m$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\leadsto$$\sf (s-b)=14m$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\leadsto$$\sf (s-c)=6m.$

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$\:$

$\therefore\:$ $\sf area\:of\:the\: whole\:plot=area\:of\:∆ABC$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:=$$\displaystyle \sqrt{s(s-a)(s-b)(s-c)}$$\sf sq\:units$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:=$$\displaystyle \sqrt{48×28×14×6}$ $\sf m²$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\underline{\boxed{\sf{=\frak{\green{336m²}}}}}$

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${\bold { \underline{\small{Area\:ungrazed}}}}$=$\sf \:whole\:area-Grazed\:area$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=$$\sf (336-77)m²$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\underline{\boxed{\sf{=\frak{\pink{259m².}}}}}$