Given:
Sides of triangle are 10 cm, 24 cm, and 26 cm.

To find:
Area and the longest altitude of the triangle.

Solution:
Step 1: Calculate the semi-perimeter of the triangle using the formula:
s = (a + b + c)/2
Where a, b, and c are the sides of the triangle.

Here, a = 10 cm, b = 24 cm, and c = 26 cm.

s = (10 + 24 + 26)/2
s = 30

Hence, the semi-perimeter of the triangle is 30 cm.

Step 2: Calculate the area of the triangle using the Heron's formula:
Area = √(s(s-a)(s-b)(s-c))

Area = √(30(30-10)(30-24)(30-26))

Area = √(30*20*6*4)

Area = √(7200)

Area = 84.85 sq. cm (approx)

Hence, the area of the triangle is 84.85 sq. cm (approx).

Step 3: Calculate the longest altitude of the triangle using the formula:
altitude = (2/Area) * base

Here, the base can be taken as any side of the triangle. Let's take the side of length 26 cm as the base.

altitude = (2/84.85) * 26

altitude = 0.61 * 26

altitude = 15.86 cm (approx)

Hence, the longest altitude of the triangle is 15.86 cm (approx).

Conclusion:
The area of the triangle is 84.85 sq. cm (approx) and the longest altitude of the triangle is 15.86 cm (approx).

120cm2..