The sides of a triangle are 35 cm, 54 cm and 61 cm, respectively. Findthe length of its longest altitude. (Take √5=2.23)

The sides of the triangle given are 35 cm, 54 cm, and 61 cm. The longest side, which is 61 cm, will be the base if we are looking for the longest altitude (since the altitude will be shortest for the longest base in a triangle).

The area of a triangle can be calculated using Heron's formula, which is suitable for this situation where all three sides of the triangle are known. Heron's formula is given by:

Area = sqrt[s(s - a)(s - b)(s - c)]

where a, b, and c are the sides of the triangle, and s is the semi-perimeter of the triangle, given by:

s = (a + b + c) / 2

First, calculate the semi-perimeter:

s = (35 cm + 54 cm + 61 cm) / 2 = 75 cm

Then, substitute the values of s, a, b, and c into Heron's formula to find the area:

Area = sqrt[75(75 - 35)(75 - 54)(75 - 61)] = sqrt[75 * 40 * 21 * 14] = 3150 cm^2

The area of a triangle can also be found by the formula:

Area = 1/2 * base * height

We can rearrange this formula to solve for the height (which is the length of the altitude we're looking for):

Height = (2 * Area) / base

Substitute the known values:

Height = (2 * 3150 cm^2) / 61 cm = 103.28 cm

So, the length of the longest altitude of the triangle is approximately 103.28 cm.