Let P, Q, R, S be four points on sides AB, BC, CD and DA respectively of a quadrilateral ABCD. Suppose PQRS is a parallelogram. Must P, Q, R and S be midpoints of the respective sides? This can be considered a possible converse question to the Midpoint Theorem for Quadrilaterals.
Step-by-step solution
Idea: To show “must” is false, one counterexample is enough. A square has so much symmetry that points equally far round each side always make a parallelogram.
- Take square ABCD with side 4 and points P on AB, Q on BC, R on CD, S on DA with AP = BQ = CR = DS = 1. Then PB = QC = RD = SA = 3.1 mark
- Triangles PBQ, QCR, RDS, SAP each have a right angle between sides 3 and 1, so they are congruent (SAS). Hence PQ = QR = RS = SP = √10.1 mark
- A quadrilateral with both pairs of opposite sides equal is a parallelogram, so PQRS is a parallelogram, yet P, Q, R, S are not midpoints. So the answer is No. (In any quadrilateral, points at the same fraction t along AB from A, along CB from C, along CD from C and along AD from A also give a parallelogram.)1 mark
Check: Coordinates: A(0, 4), B(4, 4), C(4, 0), D(0, 0); P(1, 4), Q(4, 3), R(3, 0), S(0, 1). Q − P = (3, −1) = R − S ✓, so PQRS is a parallelogram.
Answer to write in the exam
Square ABCD, side 4; AP = BQ = CR = DS = 1 ⇒ PB = QC = RD = SA = 3
∆PBQ ≅ ∆QCR ≅ ∆RDS ≅ ∆SAP (SAS: 3, 90°, 1) ⇒ PQ = QR = RS = SP
⇒ PQRS is a parallelogram (opposite sides equal)
P, Q, R, S are not midpoints
∴ No, they need not be midpoints.
Common mistakes that cost marks
- Answering “yes” because the midpoint case works; the question is whether only midpoints work.
- Giving a counterexample without proving PQRS is a parallelogram.
- Picking points where PQRS is not a parallelogram, e.g. AP = 1 but BQ = 2 in a square.
How this can come in the exam
Assertion (A): If P, Q, R, S on the sides of quadrilateral ABCD form a parallelogram, they must be the midpoints of the sides.
Reason (R): The midpoints of the sides of any quadrilateral form a parallelogram.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(D) A is false but R is true.
R is true (Midpoint Theorem for Quadrilaterals), but A is false: its converse fails, as the quarter-points of a square show.
Try one yourself
In a rectangle ABCD with AB = 8 and BC = 6, take AP = 2 on AB, BQ = 1.5 on BC, CR = 2 on CD and DS = 1.5 on DA. Is PQRS a parallelogram?
Show answer
Yes. ∆PBQ ≅ ∆RDS (PB = RD = 6, BQ = DS = 1.5, right angles) and ∆QCR ≅ ∆SAP (QC = SA = 4.5, CR = AP = 2), so PQ = RS and QR = SP: a parallelogram, though no point is a midpoint.
More questions like this
- (i) Complete the following proof of the Midpoint Theorem. Extend segment PQ beyond Q until point S. By how much should we extend PQ? It would be good to be able to prove that ∆APQ ≅ ∆CSQ. (Why?) Use this as a guide to specify the location of S and then complete this proof.
(ii) Give a similar proof of the converse of the Midpoint Theorem. Start by extending PQ up to a suitable point S. - Review all the properties of a rhombus/rectangle/square that you proved earlier. Formulate a converse of each. Decide if the converse is true. There are many possibilities here!
- Show that the sum of angles of a non-planar quadrilateral is always less than 360°. Can you find a non-planar quadrilateral ABCD for which ∠A + ∠B + ∠C + ∠D = 2°? (Hint: Think of a diagonal, say AC, as a hinge around which triangles ABC and ADC can rotate.) What happens to each angle of ABCD as you do this rotation?
- Let us see a third method to tile the plane using a 4-gon. Focus on only two coloured copies of SOME sharing a vertex. What do you see? It appears that each 4-gon is just a shifted copy of the other. Let us see exactly how. Draw two copies of SOME as shown in the figure so that the diagonals EO and EʹOʹ are collinear with O = Eʹ. Now place a cutout of one copy on top of SOME with diagonal EO drawn on it. Slide this cutout so that segment EO moves along EOʹ until EO matches EʹOʹ. Verify that your cutout exactly matches SʹOʹMʹEʹ.
- Is there a 4-gon with given side lengths?
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