1. Statement of the Midpoint Theorem
The Midpoint Theorem states that: In a triangle, the line segment joining the midpoints of any two sides is parallel to the third side and its length is half of the third side.
If \(D\) and \(E\) are midpoints of \(AB\) and \(AC\) of triangle \(ABC\), then:
\(DE \parallel BC\)
\(DE = \dfrac{1}{2} BC\)
2. Understanding the Theorem with a Diagram
Imagine triangle \(ABC\). If you mark \(D\) as the midpoint of \(AB\) and \(E\) as the midpoint of \(AC\), then connecting \(D\) and \(E\) forms a small segment that behaves like a ‘mini third side’.
This inner segment (\(DE\)) is perfectly parallel to the real third side (\(BC\)) and exactly half its length.
3. Applying the Midpoint Theorem
The midpoint theorem is very useful when solving geometry problems involving triangles, parallel lines, or quadrilaterals formed from triangles.
3.1. Example 1: Finding the Length of DE
In triangle \(ABC\), \(BC = 14\text{ cm}\). If \(D\) and \(E\) are midpoints of \(AB\) and \(AC\), then:
\(DE = \dfrac{1}{2} BC = \dfrac{1}{2} \times 14 = 7\text{ cm}\)
3.2. Example 2: Checking if a Line is a Mid-Segment
If a line inside a triangle is parallel to one side and is half its length, then the endpoints of that line lie at the midpoints of the other two sides.
This is the converse of the midpoint theorem.
4. Converse of the Midpoint Theorem
The converse states: If a line through the midpoint of one side of a triangle is parallel to the second side, then it bisects the third side.
This is useful for proving midpoint positions in constructions and proofs.
5. Using the Midpoint Theorem in Quadrilaterals
The midpoint theorem also helps in studying quadrilaterals formed by joining midpoints of sides.
5.1. Connecting Midpoints of a Quadrilateral
If you join the midpoints of all four sides of any quadrilateral, you will always form a parallelogram.
This result follows from repeated use of the midpoint theorem in triangles created by drawing a diagonal.
5.2. Mid-Segment in a Trapezium
In a trapezium, the segment joining the midpoints of the legs is parallel to the bases and its length is half the sum of the bases. This comes from applying the midpoint theorem to the triangles formed inside the trapezium.
5.2.1. Mid-Segment Formula
\(\text{Mid-segment} = \dfrac{1}{2}(a + b)\)
where \(a\) and \(b\) are the lengths of the two parallel sides.