Question

Determine whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the triangle. . .

Given side lengths:
a = 240
b = 133
c = 174

Answer 1

The area of triangle formed by given side lengths is 11317.50.

Explanation:

A triangle can be formed If the sum of the other 2 sides (except largest side) is longer than the largest side that is a+b>c

Here, given side lengths:

a = 240

b = 133

c = 174

Here, largest side is a = 240

so taking sum of other two sides,

b+c = 174+133 = 307 > 240

Hence, triangle can be formed with a, b and c as given lengths.

Calculating area using Heron's formula,

first find the semi perimeter by using given sides,

`\text{s}=\frac{\text{a+b+c}}{2}`

`\text{s}=\frac{\text{240+133+174}}{2}=273.5`

`\text{Area}=√(s(s-a)(s-b)(s-c))}`

Substitute values of s, a, b and c in above,

`\text{Area}=√(273.5(273.5-240)(273.5-133)(273.5-174))}`

`\text{Area}=√(273.5 * 33.5 * 140.5 * 99.5)}`

`\text{Area}=√(128085964.44)`

`\text{Area}=11317.50`(approx)

Thus, the area of triangle formed by given side lengths is 11317.50.

Answer 2

The triangle inequality theorem states that for a triangle to be formed, none of its side should be greater than the sum of the other two sides. The given satisfy this condition. Therefore, a triangle may be formed.
The area of the triangle is obtained by the Heron's formula,
A = sqrt ((s)(s - a)(s - b)(s - c)) ; s = (a + b + c) / 2
Plugging in the values,
s + (240 + 133 + 174) / 2 = 273.5
A = sqrt ((273.5)(273.5 - 240)(273.5 - 133)(273.5 - 174))j
A = 11317.51 unit²