Find the area of the triangle?

Question

Find the area of the triangle with vertices A(2, 1), B(5, 3), and C(6, 4).

Answer 1

Two more approaches:

Method 1. Using determinant

|1 2 1|

|1 5 3|*(1/2) =

|1 6 4|

 

|1 2 1|

|0 3 2|*(1/2) = (9-8)/2 = 1/2

|0 4 3|

 

Method 2. Using vectors

AB = (3, 2), AC = (4, 3)

area = (1/2)|AB X AC| = (1/2)|(3, 2) X (4, 3)| = (1/2)(9-8) = 1/2

 

Answer 2

There are a number of ways. Here are two methods. You will need to draw a picture first.

1. Using a bounding rectangle. You can encase this triangle in a rectangle bounded by lines x=2, x=6, y=1, and y=4. The area of this rectangle is easily calculated as 12 since it is a 4×3.

From the picture, you should see that you can get your triangle by cutting away three right triangles and a smaller rectangle. You can get their areas of the three triangles using A=bh/2. After subtracting their areas away you should get 1/2 for the answer.

2. Pick's theorem. A lattice point is any point with both coordinates being integers. Given a simple lattice polygon, if B is the number of lattice points on the perimeter, and I is the number of lattice points inside the polygon, then the polygon's area is A = B/2 + I - 1.

For your triangle you have B = 3 and I = 0 so the area is 3/2 + 0 - 1 = 1/2.