Question

the perimeter of a triangle is 36 CM if the altitude of triangle are in the ratio 1 :2:3 then find the length of the corresponding sides of the triangle (answer is coming in book is 216/11cm,108/11cm,72/11cm respectively.)​

Answer 1

Step-by-step explanation:

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

The altitudes are given in the ratio of 1:2:3.

So we can let x be such that the altitudes have lengths x,2x, and 3x.

Let x be the length of the altitude drawn to side a.

Let 2x be the length of the altitude drawn to side b.

Let 3x be the length of the altitude drawn to side c.

Now we use the usual formula for the area of a triangle, which is

Area of a triangle = 1%2F2�(any side of triangle)�(altitude drawn to that side)

Therefore

Area of the triangle = 1%2F2a�x = 1%2F2b�2x = 1%2F2�3x

1%2F2a�x = 1%2F2b�2x = 1%2F2c�3x

Multiply by 2

a�x = 2b�x = c�3x

ax = 2bx = 3cx

Divide by x

a = 2b = 3c

We are given that the perimeter is 36, so

P = a + b + c = 36

So we have the system

a = 2b

a = 3c

a + b + c = 36

Solve the first eq. for b: b = a%2F2

Solve the second eq. for c: c = a%2F3

a + a%2F2 + a%2F3 = 36

6a + 3a + 2a = 216

11a = 216

a = 216%2F11

Then b = a%2F2 = 1%2F2a = 1%2F2�216%2F11 = 108%2F11

And c = a%2F3 = 1%2F3a = 1%2F3�216%2F11 = 72%2F11

So the sides would have to be a=216%2F11, b=108%2F11, c=72%2F19

Answer 2

Answer:

your answer is in picture