4.3: Coordinate Proofs | Geometry Honors

How it works

Coordinates turn geometric claims into calculations involving slope, distance, and midpoint. Choose coordinates that simplify the figure without imposing extra conditions. Horizontal and vertical lines require separate treatment when a slope is undefined.

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\quad M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Worked examples

Example 1

Show that $\displaystyle (0,0),(4,0),(4,3),(0,3)$ form a rectangle.

Consecutive sides are horizontal and vertical, hence perpendicular. Opposite sides are parallel. Four right angles prove it is a rectangle.

Example 2

Find the midpoint of the segment from $\displaystyle (-2,4)$ to $\displaystyle (6,-2)$ .

$\displaystyle M=(({-2+6})/2,(4-2)/2)=(2,1)$ .

Problems

Problem 4.3.1

For $\displaystyle A=(0,0)$ , $\displaystyle B=(4,0)$ , $\displaystyle C=(4,3)$ , and $\displaystyle D=(0,3)$ , show the diagonals bisect each other.

Rectangle ABCD with vertices 0,0; 4,0; 4,3; 0,3 and both diagonals. — Find each diagonal midpoint using its endpoints.

Show worked solution

The midpoint of $\displaystyle AC$ is $\displaystyle (2,1.5)$ and the midpoint of $\displaystyle BD$ is also $\displaystyle (2,1.5)$ . Therefore they bisect each other.

Rectangle ABCD with vertices 0,0; 4,0; 4,3; 0,3 and both diagonals. The solution adds labeled relationships. — Both diagonal midpoints are (2, 1.5).

Problem 4.3.2

Prove $\displaystyle (0,0),(2,2),(4,0),(2,-2)$ form a rhombus.

Show worked solution

All four side lengths are $\displaystyle \sqrt{2^2+2^2}=2\sqrt{2}$ , so the quadrilateral is a rhombus.

Problem 4.3.3

Why would placing a general triangle at $\displaystyle (0,0),(a,0),(0,b)$ invalidate a proof for all triangles?

Show worked solution

It forces a right angle at the origin. A general third vertex must be allowed a nonzero horizontal coordinate.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.3.1

Problem 4.3.2

Problem 4.3.3

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