The mid-point theorem
Mark the mid-points D and E of two sides of a triangle and join them. Change the base, move the top corner and make it taller or lower: DE always stays parallel to BC and exactly half as long.
Core concept
In any triangle ABC, the line joining the mid-points D of AB and E of AC is parallel to BC and DE = ½ BC, whatever the shape of the triangle
Topic: Quadrilaterals and triangles
Take any triangle ABC. Find D, the mid-point of AB, and E, the mid-point of AC, and join them. The mid-point theorem says that DE is parallel to the third side BC and exactly half its length: DE = ½ BC. It does not matter whether the triangle is tall or flat, or whether A leans to one side. The theorem is proved with congruent triangles, and it is used to show, for example, that joining the mid-points of the sides of any quadrilateral makes a parallelogram.
What you will learn
- State the mid-point theorem.
- Show that DE = ½ BC for triangles of different shapes.
- Use the theorem to find a missing length.
Before you start
- Know what a mid-point is and what parallel lines are.
Materials
- Paper, a ruler and a pencil
Equipment
- A set square to check that DE is parallel to BC
Procedure
- Record the triangle with BC = 8 cm.
- Move A to the side and record.
- Change the height and record.
- Change BC to 10 cm and record.
How this simulation is simplified
- B and C lie on a straight line; A can move sideways and up or down.