If point L is midpoint of side AD, point M is midpoint of id BC and point N is at origin, what is the area of triangle LMN?

Question

Rectangle ABCD has vertices at A(3,7), B(7,4), C(1,-4), and D(-3,-1).

Answer 1

I think if you work along, plotting points on graph paper, what I'm saying will be much clearer.  There's a lot of visual content to the problem.

 

Let's use the midpoint formula to find L and M.

 

L = ( (3-3)/2, (7-1)/2) = (0,3)

M = ( (7+1)/2, (4-4)/2) = (4, 0)

N = (0,0) at the origin.

 

Notice that L is a straight vertical distance above the origin, and M is a straight horizontal distance to the right of the origin.  It will help to see this if you plot the points.

 

This means that segment LN is perpendicular to segment MN, and we have a right triangle.  For right triangles, the legs are the base and height in the area formula.  So the area formula is

 

A = (1/2)*base*height; using MN as base, and LN as height,

A = (1/2)*MN*LN.

 

So next, we find the lengths LN and MN.  Since these are straight vertical and horizontal distances, we don't need the distance formula.  L is 3 units directly above N, so LN = 3.  M is 4 units directly to the right of N, so MN = 4.

 

Now we plug these numbers into our formula:

 

A = (1/2)*4*3.