A method of proving geometric properties by placing figures on a coordinate plane and using algebraic formulas (distance, midpoint, slope) to verify relationships.
Instead of arguing with angles and congruence marks, drop the shape onto a grid and let algebra do the heavy lifting. Want to prove a quadrilateral is a parallelogram? Calculate all four slopes—if opposite sides have equal slopes, they're parallel, and you're done. Coordinates turn visual intuition into airtight calculation.
Showing a random 20 of 50 problems.
Example 1
easy
Find the midpoint of segment from (2a,0) to (0,2b).
Example 2
medium
Why is using variables like (a,0) and (0,b) better than specific numbers in a general coordinate proof?
Example 3
easy
Compute the slope from (3,−1) to (7,11).
Example 4
medium
Using coordinates A(0,0),B(2a,0),C(2a,2b),D(0,2b), show AB and CD are congruent.
Example 5
medium
Use a coordinate proof to show that the diagonals of a rectangle are equal in length. Place the rectangle with one vertex at the origin.
Example 6
medium
Prove the triangle (0,0),(4,0),(0,3) is right-angled at the origin via slopes.Prove the triangle is right-angled at the origin using slopes.
Example 7
challenge
Using coordinates, prove that in any triangle, the perpendicular from a vertex to the opposite side lies inside the triangle iff that vertex's angle is acute.
Example 8
medium
Using vertices (0,0), (a,0), (a,b), (0,b), what are the midpoints of the two diagonals?
Example 9
hard
Prove the right triangle with vertices (0,0),(a,0),(0,b) has its hypotenuse length a2+b2.Find the hypotenuse length using the distance formula.
Example 10
easy
In a general coordinate proof for any rectangle, why choose vertices (0,0), (a,0), (a,b), (0,b)?
Example 11
medium
Use a coordinate proof to show that the segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side (Midsegment Theorem). Use triangle with vertices A(0,0), B(2a,0), C(2b,2c).
Example 12
medium
Using (0,0),(a,0),(a,b),(0,b), show the diagonals of any rectangle are equal.
Example 13
medium
Prove (0,0),(4,0),(4,4),(0,4) is a rhombus by showing all four sides are equal.Show all four sides are equal using the distance formula.
Example 14
hard
Using coordinates (0,0),(a,0),(a,b),(0,b), prove the diagonals of any rectangle bisect each other.
Example 15
challenge
Prove that the diagonals of a square are perpendicular using (0,0),(a,0),(a,a),(0,a).
Example 16
medium
Show that the midpoints of the sides of triangle (0,0),(4,0),(0,6) form a triangle with half the perimeter.Midpoints of the sides form the medial triangle (dashed). Show its perimeter is half the original.
Example 17
challenge
Using coordinates, prove that the three medians of triangle (0,0),(6,0),(3,6) are concurrent at the centroid.
Example 18
medium
For the triangle A(0,0),B(6,0),C(2,4), is the median from C to midpoint of AB perpendicular to AB?Is the median from C to midpoint M of AB perpendicular to AB?
Example 19
medium
Show that the diagonals of the square with vertices (0,0),(a,0),(a,a),(0,a) have equal length.
Example 20
easy
When setting up a coordinate proof, why place one vertex at the origin?