ANSWER:

The area of the triangle is 720 cm^2

The altitude of triangle corresponding to the shortest side is 80 cm

Step-by-step explanation:

To find the area of the triangle and the altitude corresponding to the shortest side

PERIMETER = 180 cm(given)

Two sides are 18 cm and 80 cm(given)

Step 1 = Find the third side of the triangle through the perimeter of the triangle by naming the side as x

That is,

18 + 80 + x = 180

180 - 98 = x

x = 82 cm

Step 2 = As we know that the area of the triangle when a, b and c are three different sides  is

Area = s(s - a)(s - b)(s - c) WHOLE ROOT

s = perimeter/2

 = 180/2

s = 90 cm

a = 18 cm

b = 80 cm

c = 82 cm

Now find the area

Area = 90(90 - 18)(90 - 80)(90 - 82) Whole root

         = 90(72)(10)(8) Whole root

          = 9 X 10[90] X 2 X 36[72] X 10 X 2 X 4[8] Whole root

         = 9 X 10 X 10 X 2 X 2 X 36 X 4 Whole root[rearranged]

         = 9 X 100 X 4 X 36 X 4 Whole root

As the numbers are in the square root so we will find square of each number

= 3[3^2 = 9] X 10[10^2 = 100] X 2[2^2 = 4] X 6[6^2 = 36] X 2[2^2 = 4]

The sign of the square root of the numbers will be removed as we found the square roots of the numbers

Now,

Area = 3 X 10 X 2 X 6 X 2

        = 30 X 24

Area = 720 cm^2

Step 3 = Now equate the two different ways to calculate the areas of the triangle

As, according to the question we have to find altitude corresponding to the shortest side

That is 18 cm

Equation forms like this

Area = Area(of the triangle)

s(s - a)(s - b)(s - c) whole root = 1/2 X base X height

720 cm^2 = 1/2 X 18 X HEIGHT

Find the value of the altitude corresponding to the shortest side through the rules of linear equation.

720 = 9 X HEIGHT

720/9 = HEIGHT

Altitude(height) = 80 cm