Counterexamples

What is a counterexample?

A counterexample is an example in which the hypothesis is true, but the conclusion is false. If you can find a counterexample to a conditional statement, then that conditional statement is false. Using counterexamples to prove that a property or a theorem don't hold is sometimes easier that writing a complete proof. We will explore a couple of interactive examples below.

Congruence of triangles and union

Even if you never heard about the concept of congruence, you can easily recognize two congruent geometric objects, because they are identical in size and shape. Use the applet below to create a counterexample for the following statement: If a triangle is congruent to a triangle (and we write this as ) and a triangle is congruent to a triangle (), then the union of and () is congruent to the union of and (). You can drag the triangles by dragging the whole shape, or using the red points, and rotate them using the green points.

Rectangles with given area

Explore the applet below that shows (how many?) counterexamples for the following statement: If a rectangle has area 16, then its perimeter is 20. Can you also show that there is a special case in which the rectangle is actually a square?