Question

7. The perimeter of a triangular field is 240 dm. If two of its sides are 78 dm and 50 dm, find
the length of the perpendicular on the side of length 50 dm from the opposite vertex​

Answer 1

$\displaystyle\large\underline{\sf\red{Given}}$

✭ Perimeter of a triangular field is 240 dm

✭ Two of it's sides are of length 70 dm & 50 dm

$\displaystyle\large\underline{\sf\blue{To \ Find}}$

◈ The length of the perpendicular from the opposite vertex on the 50 dm side?

$\displaystyle\large\underline{\sf\gray{Solution}}$

So hee we shall first find the area with the help of the heron's formula and then find the height with the help of the formula to find area of a triangle!!

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$\underline{\bigstar\:\textsf{According to the given Question :}}$

So first we shall find the semi Perimeter of the triangle,

›› $\displaystyle\sf Semi \ Perimeter = \dfrac{Perimeter}{2}$

›› $\displaystyle\sf Semi \ Perimeter = \dfrac{240}{2}$

›› $\displaystyle\sf \green{Semi \ Perimeter = 120}$

So then the Third side will be,

›› $\displaystyle\sf Perimeter = Sum \ of \ all \ sides$

›› $\displaystyle\sf 240 = 78+50+x$

›› $\displaystyle\sf 240 = 128+x$

›› $\displaystyle\sf 240-128 = x$

›› $\displaystyle\sf \purple{Third \ side = 112 \ dm}$

So then using Heron's formula,

$\displaystyle\underline{\boxed{\sf Heron's \ Formula = \sqrt{s(s-a)(s-b)(s-c)}}}$

• s = 120
• a = 78
• b = 50
• c = 112

Substituting the values,

»» $\displaystyle\sf Heron's \ Formula = \sqrt{s(s-a)(s-b)(s-c)}$

»» $\displaystyle\sf \sqrt{120(120-78)(120-50)(120-112)}$

»» $\displaystyle\sf \sqrt{120\times 70\times 42\times 8}$

»» $\displaystyle\sf \sqrt{2822400}$

»» $\displaystyle\sf \orange{Area = 1680 \ dm^2}$

Now we also know another Formula to find the area of a triangle, that is,

$\displaystyle\underline{\boxed{\sf Area = \dfrac{1}{2}\times Base \times Height}}$

• Base = 50 dm
• Area = 1680 dm²

Substituting the values,

➳ $\displaystyle\sf Area = \dfrac{1}{2}\times Base \times Height$

➳ $\displaystyle\sf 1680 = \dfrac{1}{2} \times 50 \times Height$

➳ $\displaystyle\sf 1680\times 2 = 50\times Height$

➳ $\displaystyle\sf 3360 = 50\times Height$

➳ $\displaystyle\sf \dfrac{3360}{50} = Height$

➳ $\displaystyle\sf \pink{Height = 67.2 \ dm}$

$\displaystyle\therefore\:\underline{\sf Length \ of \ the \ perpendicular \ is 67.2dm}$

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