Question

${\color{yellow} \bigstar} \: \underline \color{magenta} \mathbb{QUESTION :-}$

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

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Answer 1

Answer:

∴ Ungrazed Area of Plot will be 259 m²

Step-by-step explanation:

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,

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

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