Question

Area of a triangle are in the ratio 5 :12 :13 and its perimeter is 150m. Find its area

Answer 1

Answer:

Area=750m

Step-by-step explanation:

5x+12x+13x=150m

30x=150m

x=150/30

x=5

Side a=5x

=5*5

=25

Side b=12*5

=60

Side c=13*5

=65

Using Heron's formula-

Area=750m

Answer 2

Correct Question :

• The sides of a triangle are in the ratio 5 : 12 : 13 and its perimeter is 150 m. Find its area.

Given :

• Sides of a triangle are in the ratio 5 : 12 : 13.
• Perimeter is 150 m.

To Find :

• Area of the triangle

Solution :

Let's assume the sides of the triangle as :

$\bullet \: \: \rm 5x, \: 12x \: and \: 13x$

Now, let's find the perimeter of the sides,

• Perimeter of triangle = Sum of all sides

$\implies \rm 5x + 12x + 13x \: = 150 \\ \\ \\ \implies \rm \: 30x = 150 \\ \\ \\ \implies \rm \: x = \cancel \dfrac{150}{30} \\ \\ \\ \implies \: \boxed {\rm{x = 5}}$

Hence, perimeter of the triangle is 5.

Now, let's calculate the sides of the triangle, we'll be finding the sides of triangle by multiplying the sides with the perimeter, i.e.

$\tiny\dag\tiny \: \bf { \underline{Calculating \: for \: first \: side :}}$

$\implies \rm 5x \: = 5 \times 5 \\ \\ \\ \implies \bf 25$

$\tiny\dag\tiny \: \bf { \underline{Calculating \: for \: second \: side :}}$

$\implies \rm 12x \: = 12 \times 5 \\ \\ \\ \implies \bf 60$

$\tiny\dag\tiny \: \bf { \underline{Calculating \: for \: third \: side :}}$

$\implies \rm 13x \: = 13 \times 5 \\ \\ \\ \implies \bf 65$

Therefore, the sides are :-

• 25 m
• 60 m
• 65 m

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Now, let us calculate the semi-perimeter of the triangle,

Formula used :

We'll be using heron's formula (for triangle) to calculate the semi-perimeter of the triangle :-

• $\tt \: S \: = \dfrac{a + b + c}{2}$

Where,

• a = 25 m
• b = 60 m
• c = 65 m

$\dag$ Putting the values,

$\implies \rm S = \dfrac{a + b + c}{2} \\ \\ \\ \implies \rm S = \dfrac{25 + 60 + 65}{2} \\ \\ \\ \implies \rm S = \cancel \dfrac{150}{2} \\ \\ \\ \implies \bf S \: = \: 75$

Therefore, semi-perimeter of triangle is 75 m.

⠀⠀__________________

According to the Question,

Now, let's calculate the area of triangle.

Formula to be used :

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

Putting the values,

$\implies \rm \sqrt{75(s - 25)(s - 60)(s - 65)} \\ \\ \\ \implies \rm \sqrt{75(75 - 25)(75 - 60)(75 - 65)} \\ \\ \\ \implies \rm \sqrt{75 \times 50 \times 15 \times 10} \: \\ \\ \\ \implies \rm \sqrt{562500} \\ \\ \\ \implies \bf 750 \: m^2$

Therefore, area of the triangle is 750 m².