This is a fundamental lesson in geometry that creates a powerful link between the sides of a triangle. It's often used in exam questions to prove relationships between lines and calculate lengths. The lesson is built around two key theorems: the theorem itself and its converse.

1. The Midpoint Theorem

"The straight line segment joining the midpoints of two sides of a triangle is parallel to the third side and is equal in length to half of it."

Let's break this down into two simple parts. In triangle ABC, if:

• P is the midpoint of AB

• Q is the midpoint of AC

Then two things are always true:

1. Parallelism: PQ is parallel to BC (PQ || BC).

2. Length: The length of PQ is exactly half the length of BC (PQ = ½ BC).

Exam Application (Calculations):

• If you know the length of PQ, you can find BC. (e.g., if PQ = 5 cm, then BC = 10 cm).

• If you know the length of BC, you can find PQ. (e.g., if BC = 12 cm, then PQ = 6 cm).

2. The Converse of the Midpoint Theorem

The "converse" is simply the reverse of the theorem. It starts with different information and arrives at a different conclusion.

"The straight line drawn through the midpoint of one side of a triangle, parallel to another side, bisects the third side."

Let's break this down. In triangle ABC, if:

• P is the midpoint of AB

• The line PQ is drawn parallel to BC (PQ || BC)

Then the conclusion is:

• Q must be the midpoint of the third side, AC. (AQ = QC).

How to Write Proofs (Riders) for Exams

Using these theorems in proofs requires a clear, step-by-step structure.

Using the Midpoint Theorem:

1. State the triangle: "In triangle ABC..."

2. State the knowns (the midpoints): "...P is the midpoint of AB and Q is the midpoint of AC (Given)."

3. State the conclusion and the theorem: "Therefore, PQ || BC (Midpoint Theorem)."

Using the Converse of the Midpoint Theorem:

1. State the triangle: "In triangle ABC..."

2. State the knowns (one midpoint, one parallel line): "...P is the midpoint of AB (Given) and PQ || BC (Given)."

3. State the conclusion and the theorem: "Therefore, Q is the midpoint of AC (Converse of the Midpoint Theorem)."

Exam Tips & Common Problems

• Tip 1: Look for the Clues! In a geometry problem, if you see the word "midpoint" or see tick marks on a diagram indicating a side is bisected (like | on AP and PB), your brain should immediately think of the Midpoint Theorem.

• Common Problem: The Quadrilateral Proof. A very frequent exam question is: "Prove that the quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram."

  • How to solve it:

    1. Draw the quadrilateral (e.g., ABCD) and its midpoints (P, Q, R, S).

    2. Draw a diagonal (e.g., AC). This splits the quadrilateral into two triangles (ΔABC and ΔADC).

    3. Apply the Midpoint Theorem to each triangle separately:

       • In ΔABC, PQ || AC and PQ = ½ AC.

       • In ΔADC, SR || AC and SR = ½ AC.

    4. Combine the results: Since both PQ and SR are parallel to AC and equal to half its length, they must be parallel to each other and equal in length (PQ || SR and PQ = SR).

    5. Conclusion: "Since one pair of opposite sides (PQ and SR) is both equal and parallel, PQRS is a parallelogram."

• Mistake to Avoid: Don't mix up the theorem and its converse.

  • Theorem: Starts with 2 midpoints -> Proves parallel and half length.

  • Converse: Starts with 1 midpoint + 1 parallel line -> Proves the line creates a second midpoint.