Midpoints and coordinate measurements

The midpoint averages corresponding coordinates: \(M=\left(\dfrac{x_A+x_B}{2},\dfrac{y_A+y_B}{2}\right)\). This follows because the horizontal and vertical changes are each halved. For area, identify a base and its perpendicular height from the coordinates.

Example

For \(A(-2,1)\) and \(B(4,1)\), the midpoint is \((1,1)\), and the horizontal length \(AB\) is \(6\). With \(C(1,5)\), the perpendicular height above \(AB\) is \(4\), so the triangle area is \(\dfrac12(6)(4)=12\) square units.

A sketch suggests relationships; calculations establish them. A sloping side is not the perpendicular height, and one right angle alone does not establish that a quadrilateral is a rectangle.