Question

The sides of a triangle are 35 cm, 54 cm and 61 cm, respectively. The length of its longest altitude ______.

Options

• `16sqrt(5)` cm

• `10sqrt(5)` cm

• `24sqrt(5)` cm

• 28 cm

Answer 1

The sides of a triangle are 35 cm, 54 cm and 61 cm, respectively. The length of its longest altitude `underlinebb(24sqrt(5)  cm)`.

Explanation: 

Given: The sides of a triangle area = 35 cm, b = 54 cm and c = 61 cm, respectively.

So, semi-perimeter of a triangle is:

`s = (a + b + c)/2`

= `(35 + 54 + 61)/2`

= `150/2`

= 75

Area of triangle = `sqrt(s(s - a)(s - b)(s - c))`

= `sqrt(75(75 - 35)(75 - 54)(75 - 61))`

= `sqrt(75 xx 40 xx 21 xx 14)`

= `sqrt(5 xx 5 xx 3 xx 2 xx 2 xx 2 xx 5 xx 3 xx 7 xx 7 xx 2)`

= `5 xx 3 xx 2 xx 2 xx 7sqrt(5)`

= `420sqrt(5)`

As know that, 

Area of triangle ABC = `1/2` × Base × Altitude

`1/2` × 35 × Altitude = `420sqrt(5)`

Altitude = `(420sqrt(5) xx 2)/35`

Altitude = `24sqrt(5)`

Therefore, the length of altitude is `24sqrt(5)`.