Question

Find the area of a triangle whose sides are 50m , 78m , 112m , respectively and also find the perpendicular from the opposite angle on the side 112 m .

Answer 1

$<b><u><i>Heya mate!!! Here's your answer</i></u>$
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$\huge{Given}$

The sides of a triangle are 50 m, 78 m, and 112 m respectively.

$\huge{To\: Find}$

(1) The area of the triangle.

(2) The length of perpendicular corresponding to the side with base 112 m

$\huge{Solution}$

$\mathcal{We\: divide \: answer\: in\: two\: parts}$

$\bf{\pink{First \:part \: -}}$

The sides of the triangle as given are - 50 m, 78 m and 112 m

Let the first side be 'a'
Second side be 'b'
Third side = 'c'

Then,
a = 50 m

b = 78 m

c = 112 m
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Then,

Perimeter (P) = a + b + c

= 50 + 78 + 112 m

= 240 m

And,

Semi perimeter (s) = perimeter/2

= 240/2

= 120 m
_______________________

Now, by the heron's formula, which is used to find the area of a triangle, we get,

Area = √[s(s-a)(s-b)(s-c)]

= √[120(120-50)(120-78)(120-112)]

= √[120(70)(42)(8)]

= √[2×2×2×3×5(2×5×7)(2×3×7)(2×2×2)]

=√[2×2×2×2×2×2×2×2×3×3×5×5×7×7]

= 2×2×2×2×3×5×7 m²

= 1680 m². [Equation 1]

That's the first answer.
_________________________

$\bf{\pink{Second\: part\: -}}$

Now, base = 112 m

Let the height be 'h'

We know that the area of any triangle is equal to half the product of its base and corresponding height (perpendicular).

Hence, the area of triangle

= ½ × 112 × h

= 56 h

But, by Equation 1, the area is 1680 m².

Hence, both areas are equal.

=> 56h = 1680

=> h = 1680/56

=> h = 30 m. [Equation 2]

That is your second answer.
________________________

$\mathcal{\pink{Finally,}}$

From equations 1 & 2, your answers are,

$\boxed{\bold{\blue{\mathcal{(1)\: = \: 1680 \: m^2}}}}$
$\boxed{\bold{\green{\mathcal{(2)\: = \: 30 \: m}}}}$
______________________________

$\huge{\bold{\red{\mathfrak{Thank\: you}}}}$

Answer 2

The sides of a triangle are 50 m, 78 m, and 112 m respectively.

(1) The area of the triangle.

(2) The length of perpendicular corresponding to the side with base 112 m

The sides of the triangle as given are - 50 m, 78 m and 112 m

Let the first side be 'a'

Second side be 'b'

Third side = 'c'

Then,

a = 50 m

b = 78 m

c = 112 m

________________________

Then,

Perimeter (P) = a + b + c

= 50 + 78 + 112 m

= 240 m

And,

Semi perimeter (s) = perimeter/2

= 240/2

= 120 m

_______________________

Now, by the heron's formula, which is used to find the area of a triangle, we get,

Area = √[s(s-a)(s-b)(s-c)]

= √[120(120-50)(120-78)(120-112)]

= √[120(70)(42)(8)]

= √[2×2×2×3×5(2×5×7)(2×3×7)(2×2×2)]

=√[2×2×2×2×2×2×2×2×3×3×5×5×7×7]

= 2×2×2×2×3×5×7 m²

= 1680 m². [Equation 1]

That's the first answer.

_________________________

Now, base = 112 m

Let the height be 'h'

We know that the area of any triangle is equal to half the product of its base and corresponding height (perpendicular).

Hence, the area of triangle

= ½ × 112 × h

= 56 h

But, by Equation 1, the area is 1680 m².

Hence, both areas are equal.

=> 56h = 1680

=> h = 1680/56

=> h = 30 m. [Equation 2]

That is your second answer.

________________________

From equations 1 & 2, your answers are,

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