Question

Find area of triangle formed by (1,1) (2,3) (4 ,5) using heron's formula ​

Answer 1

The area of a triangle is:

• A = 1 unit²

Explanation:

Given the points of a triangle

A(1, 1)

B(2, 3)

C(4, 5)

`\mathrm{Compute\:the\:distance\:between\:}\left(x_1,\:y_1\right),\:\left(x_2,\:y_2\right):\quad √(\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2)`

Distance between AB:

`AB=√(\left(2-1\right)^2+\left(3-1\right)^2)`

`=√(1+4)`

`=√(5)`

so

• 
  `a=√(5)`

Distance between BC:

`BC=√(\left(4-2\right)^2+\left(5-3\right)^2)`

`=√(2^2+2^2)`

`=√(2^3)`

`=2√(2)`

so

• 
  `b=2√(2)`

Distance between AC:

`AC=√(\left(4-1\right)^2+\left(5-1\right)^2)`

`=√(3^2+4^2)`

`=√(5^2)`

`=5`

so

• 
  `c=5`

Semiperimeter = sum of sides ÷ 2

`s=(√(5)+2√(2)+5)/(2)=5.03224`

Area of the triangle:

`A=√(s\left(s-a\right)\left(s-b\right)\left(s-c\right))`

• 
  `a=√(5)`

• 
  `b=2√(2)`

• 
  `c=5`

`A=\sqrt{s\left(s-√(5)\right)\left(s-2√(2)\right)\left(s-5\right)}`

`A=\sqrt{5.03224\dots \left(5.03224\dots -√(5)\right)\left(5.03224\dots -2√(2)\right)\left(5.03224\dots -5\right)}`

`A=1`

Therefore, the area of a triangle is:

• A = 1 unit²