Introduction: This is a powerful theorem in geometry. The key is to correctly identify the triangle you are working with and which sides the midpoints are on. The converse is just as important. Expect "rider" questions that require you to prove properties of quadrilaterals.

The "S Pass" Foundation (නියත S එකකට)

• Prompt 1 (State): State the Midpoint Theorem, referring to a triangle ABC with midpoints P on AB and Q on AC.

• Prompt 2 (State): State the converse of the Midpoint Theorem.

• Prompt 3 (Apply): In ΔPQR, X and Y are midpoints of PQ and PR. If QR = 18 cm, find the length of XY.

• Prompt 4 (Apply): In ΔABC, D is the midpoint of AB. A line from D parallel to BC meets AC at E. If AC = 10 cm, what is the length of AE?

• Prompt 5 (Identify): The sides of a triangle are 6 cm, 8 cm, and 10 cm. Find the perimeter of the triangle formed by joining the midpoints of the sides.

Climbing to a "C" (C එකට පාර)

• Prompt 1 (Problem Solve): In quadrilateral ABCD, P, Q, R, S are the midpoints of the sides. If diagonal AC = 12 cm and diagonal BD = 15 cm, find the perimeter of PQRS.

• Prompt 2 (Proof): In ΔABC, P and Q are midpoints of AB and AC. Prove that the quadrilateral PBCQ is a trapezium.

• Prompt 3 (Application): In ΔLMN, P is the midpoint of LM. The line through P parallel to MN meets LN at Q. The line through Q parallel to LM meets MN at R. Prove that LM = 2QR.

• Prompt 4 (Proof): Prove that the line segment joining the midpoints of two sides of a triangle divides it into a smaller triangle and a trapezium.

• Prompt 5 (Problem Solve): In right-angled triangle ABC, B^=90∘. P is the midpoint of the hypotenuse AC. A line through P parallel to BC meets AB at Q. Prove that Q is the midpoint of AB and PQ=21​BC.

Aiming for a "B" (B ඉලක්කය)

• Prompt 1 (Rider): Prove that the quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram. (This is a classic proof).

• Prompt 2 (Rider): In trapezium ABCD, AB || DC and P is the midpoint of AD. A line through P parallel to AB meets BC at Q. Prove that Q is the midpoint of BC.

• Prompt 3 (Rider): Prove that the four triangles formed by joining the midpoints of the three sides of a triangle are congruent to each other.

• Prompt 4 (Deduction): In ΔABC, D and E are the midpoints of AB and AC. The median from A meets BC at F. Prove that the line DE bisects the median AF.

• Prompt 5 (Rider): Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

Securing the "A" Distinction (A සාමාර්ථය තහවුරු කරගන්න)

• Prompt 1 (Challenge Rider): Prove that the quadrilateral formed by joining the midpoints of the sides of a rhombus is a rectangle.

• Prompt 2 (Challenge Rider): Prove that the quadrilateral formed by joining the midpoints of the sides of a rectangle is a rhombus.

• Prompt 3 (Multi-step Rider): In ΔABC, D is the midpoint of BC. P is the midpoint of AD. The line BP produced meets AC at Q. Prove that AQ=31​AC. (Hint: Draw a line through D parallel to BQ).

• Prompt 4 (Synthesis): ABCD is a parallelogram. E is the midpoint of BC. The line DE intersects the diagonal AC at F. Prove that F is a point of trisection of AC (i.e., prove AF=31​AC or FC=32​AC).

• Prompt 5 (Formal Proof): Provide a formal proof for the Midpoint Theorem using congruent triangles and properties of parallelograms.