Question

sides of a triangle are in the ratio of 5:7:10 and its perimeter is 40 cm find its area​ ​

Answer 1

Answer:

perimeter of triangle = 40 cm

tri in the ratio = 5+7+10= 22 cm

area

of tri = 1/2 * b * h

=

1/2 *40 *22

=440

Answer 2

Given :-

Ratio of the sides of the triangle = 5 : 7 : 10

Perimeter of the triangle = 40 cm

To Find :-

The area of the triangle.

Analysis :-

Consider the common ratio as a variable.

Make an equation accordingly such that the three sides of the triangle is multiplied to the variable each added is equal to the perimeter of the triangle.

Find the value of the variable and substitute them in the sides and find them accordingly.

Next, we should find the semi perimeter by dividing the perimeter given by two.

Finally, substitute the values we got using Heron's formula and find the area accordingly.

Solution :-

We know that,

• p = Perimeter
• a = Area
• s = Semi perimeter

Let the common ratio be 'x'. Then the sides would be 5x. 7x and 10x.

Given that,

Perimeter (p) = 40 cm

Making an equation,

5x + 7x + 10x = 40

20x = 40

By transposing,

x = 40/20

x = 2

Finding the sides,

5x = 5 × 2 = 10 cm

7x = 7 × 2 = 14 cm

8x = 8 × 2 = 16 cm

Therefore, the sides of the triangle are 10 cm, 14 cm and 16 cm.

Finding the semi perimeter,

$\underline{\boxed{\sf Semi \ perimeter=\dfrac{a+b+c}{2} }}$

Given that,

Perimeter (p) = 40 cm

First side (a) = 10 cm

Second side (b) = 14 cm

Third side (c) = 16 cm

Substituting their values,

10+14+16/2 = 40/2

= 20 cm

Therefore, the semi perimeter of the triangle is 20 cm.

Using Heron's formula,

$\underline{\boxed{\sf Area \ of \ triangle=\sqrt{s(s-a)(s-a)(s-b)(s-c)} }}$

Given that,

Semi perimeter (p) = 20 cm

First side = 10 cm

Second side = 14 cm

Third side = 16 cm

Substituting their values,

$\sf =\sqrt{20(20-10)(20-14)(20-16)}$

$\sf =\sqrt{20 \times 10 \times 6 \times 4 }$

$\sf =\sqrt{4800 }$

$\sf =69.2 \ m^2$

Therefore, area of the triangle is 69.2 m².