Question

Two sides of a triangular field are 85 m and 154 m in length and its
perimeter is 324 m. Find (i) the area of the field and (ii) the length of the
perpendicular from the opposite vertex on the side measuring 154 m.​

Answer 1

Answer:

Area of Triangle=2772m^2

Length of perpendicular=1296m

Step-by-step explanation:

Let the sides of the triangle be 'a' , 'b' and'c'

a=85m

b=154m

perimeter of triangle=a+b+c

324=85+154+c

c=85

By Heron's formula,

$area \: of \: triangle = \sqrt{s(s - a)(s - b)(s - c)}$

$s = \frac{a + b + c}{2}$

$s = \frac{85 + 154 + 85}{2} = \frac{324}{2} = 162$

$(s - a) = 162 - 85 = 77$

$(s - b) = 162 - 154 = 8$

$(s - c) = 162 - 85 = 77$

$area = \sqrt{162 \times 77 \times 8 \times 77 } = 77 \sqrt{162 \times 8} = 77 \sqrt{1296} = 77 \times 36 = 2772 {m}^{2}$

Because we have an isosceles triangle here , perpendicular from The vertex joining equal sides ,divides the opposite side in two equal halves. Let the length of perpendicular bisector be equal to 'x'

Therefore By Pythagoras theorem

${85}^{2} = { (\frac{154}{2} )}^{2} + {x}^{2}$

$x = {85}^{2} - {77}^{2}$

$x = (85 - 77) \times (85 + 77)$

$x = 8 \times 162$

$x = 1296$

So length of perpendicular =1296m

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