Question

Choose ALL that correctly identify the area of the triangle given the coordinates of its vertices.
A(1, 2), B(4, 5), C(4, 0) area = 9.5 units2
A(1, 1), B(5, 2), C(3, 5) area = 3 units2
A(−3, 1), B(1, 3), C(2, −4) area = 12 units2
A(−3, −1), B(−5, −4), C(−2, −5) area = 5.5 units2

Answer 1

We have to identify the option which correctly identifies the area of triangle.

Area of triangle with coordinates $(x_{1},y_{1})$, $(x_{2},y_{2})$ and $(x_{3},y_{3})$ is given by the formula:

$\frac{1}{2}[x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})]$

1. Consider the first option,

A(1, 2), B(4, 5), C(4, 0)

Area = $\frac{1}{2}[1(5-0)+4(0-2)+4(2-5)]$

= 7.5 $units^2$

2. Consider the second option,

A(1, 1), B(5, 2), C(3, 5)

Area = $\frac{1}{2}[1(2-5)+5(5-1)+3(1-2)]$

= 7 $units^2$

3. Consider the third option,

A(−3, 1), B(1, 3), C(2, −4)

Area = $\frac{1}{2}[-3(3+4)+1(-4-1)+2(1-3)]$

= 15 $units^2$

4. Consider the fourth option,

A(−3, −1), B(−5, −4), C(−2, −5)

Area = $\frac{1}{2}[-3(-4+5)-5(-5+1)-2(-1+4)]$

= 5.5 $units^2$

So, option 4 is the correct answer.