Question

The perimeter of triangular field is 450m and its sides are in the ratio 13:12:5.Find the area of the triangle.​

Answer 1

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Answer 2

Given :–

• Perimeter of a triangular field =450m.
• Sides of a triangular field is in ratio = 13 : 12 : 5.

To Find :–

• Area of a triangular field.

Solution :–

Let,

The first side be 13x.

The second side be 12x.

The third side be 5x.

First, we need to find the value of all the three sides.

According to the question,

$\longmapsto13x + 12x + 5x = 450m$

$\longmapsto13x + 17x = 450m$

$\longmapsto30x = 450m$

$\longmapsto x = \dfrac{45 \cancel0}{ 3\cancel0} m$

$\longmapsto x = \cancel\dfrac{45}{3}m$

$\longmapsto x = 15m$

So, the angles are,

First angle = 13x = 13 × 15m = 195m

Second angle = 12x = 12 × 15m = 180m

Third angle = 5x = 5 × 15m = 75m

Now, we have to find the area of a triangular field.

By heron's formula,

$\bullet \: \blue{ \boxed {\pink{{ \sqrt{s(s - a)(s - b)(s - c)} }}}} \: \bullet$

Where,

$\boxed{s = \dfrac{perimeter}{2} }$

Given that,

• Perimeter = 450m.

$\longmapsto s = \cancel \dfrac{450}{2}m$

$\longmapsto s = 225m$

Now,

We have,

• s = 225m.
• a = 195m.
• b = 180m.
• c = 75m.

$\longmapsto\sqrt{225(225 - 195)(225 - 180)(225 - 75) } \: {m}^{2}$

$\longmapsto \sqrt{225(30)(45)(150) } \: {m}^{2}$

$\longmapsto \sqrt{(5 \times 5 \times 3 \times 3)(2 \times 3 \times 5)(3 \times 3 \times 5)(2 \times 3 \times 5 \times 5)} \: {m}^{2}$

$\longmapsto \sqrt{5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 2 \times 2} \: {m}^{2}$

$\longmapsto\sqrt{ {(5)}^{2} \times {(5)}^{2} \times {(5)}^{2} \times {(3)}^{2} \times {(3)}^{2} \times {(3)}^{2} \times {(2)}^{2} } \: {m}^{2}$

$\longmapsto5 \times 5 \times 5 \times 3 \times 3 \times 3 \times 2 \: {m}^{2}$

$\longmapsto6750 \: {m}^{2}$

Hence,

The area of a triangular field is 6750 m².