# A triangle has two sides of lengths 8 and 6 cm. Find the length of the third side if its area is 24 cm^2.

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Here we need to find the length of the third side given that the length of two sides is 8 and 6 and the area of the triangle is 24 cm^2.

Now the area of a triangle is given by Area = (1/2)*base*height

Here the area is 24 which we see is also equal to (1/2)*8*6. Therefore the base and the height of the triangle are 8 and 6. Therefore we know this is a right triangle with the smaller sides being 8 and 6. The value of the hypotenuse can be found by finding the square root of the sum of the square of 6 and 8.

8^2 + 6^2 = 64 + 36 = 100

The hypotenuse is sqrt 100 = 10.

**The required length of the third side is 10.**

The first answer works for a right triangle only.

The second answer 24*2/(8*6) = 1. So how can that work?

The area A of the triangle is given by :

(1/2)ab sinC = 24, where a and b are the sides and C is the angle between the sides.

a = 8 and b = 6 and area = 24.

Therefore (1/2)8*6sinC = 24.

sinC = 24*2/(8*6) = 1/2.

Therefore C = 30 degree, or 150 deg.

If C = 30, c ^2 = (a^2+b^2- 2abcosC)

c = sqrt(8^2+6^2 - 2*8*6 cos30) = 4.1063 cm. nearly.

If C= 150 degree, then

c = sqrt(8^2+6^2- 2*8*6*cos150) = 13.5329cm nearly.