The midpoints of the four sides of a quadrilateral are the vertices of a parallelogram.
Step-by-step solution
To find: Prove that PQRS is a parallelogram
Idea: Each side of PQRS joins the midpoints of two sides of a triangle cut off by a diagonal, so it is parallel to (and half of) that diagonal.
- Draw diagonal AC. In ∆ABC, P and Q are midpoints of AB and BC, so PQ ‖ AC and PQ = ½AC. In ∆ADC, S and R are midpoints of AD and DC, so SR ‖ AC and SR = ½AC.1 mark
- Hence PQ ‖ SR (both parallel to AC). In the same way, with diagonal BD: in ∆BCD, QR ‖ BD; in ∆BAD, PS ‖ BD; so QR ‖ PS.1 mark
- Both pairs of opposite sides of PQRS are parallel, so PQRS is a parallelogram. (Also PQ = SR = ½AC, which alone proves it.)1 mark
Check: A(0, 4), B(6, 6), C(7, 0), D(−1, 1): P(3, 5), Q(6.5, 3), R(3, 0.5), S(−0.5, 2.5). PQ = (3.5, −2) and SR = (3.5, −2): equal and parallel ✓.
Answer to write in the exam
In ∆ABC: PQ ‖ AC, PQ = ½AC (Midpoint Theorem)
In ∆ADC: SR ‖ AC, SR = ½AC (Midpoint Theorem)
⇒ PQ ‖ SR (and PQ = SR)
In ∆BCD: QR ‖ BD; in ∆BAD: PS ‖ BD ⇒ QR ‖ PS
∴ PQRS is a parallelogram.
Common mistakes that cost marks
- Using the same diagonal for all four sides. PQ and SR use AC; QR and PS use BD.
- Concluding a rectangle or rhombus. In general PQRS is only a parallelogram; it is a rhombus when AC = BD and a rectangle when AC ⟂ BD.
- Joining the midpoints in the wrong order (P to R) and getting crossing segments.
How this can come in the exam
The diagonals of quadrilateral ABCD are 10 cm and 16 cm. The perimeter of the quadrilateral formed by joining the midpoints of its sides is
- 13 cm
- 26 cm
- 52 cm
- 20 cm
Show answer
(B) 26 cm
Its sides are ½ × 10 = 5 cm and ½ × 16 = 8 cm (two of each), so the perimeter is 2(5 + 8) = 26 cm.
P, Q, R, S are the midpoints of the sides AB, BC, CD, DA of a rectangle ABCD. Show that PQRS is a rhombus.
Show answer
PQ = ½AC, QR = ½BD, RS = ½AC, SP = ½BD (Midpoint Theorem in four triangles) (1 mark). The diagonals of a rectangle are equal, AC = BD, so all four sides of PQRS are equal: a rhombus (1 mark).Try one yourself
If the diagonals of quadrilateral ABCD are perpendicular, what kind of parallelogram is formed by joining the midpoints of its sides?
Show answer
A rectangle: its sides are parallel to AC and BD, which are perpendicular, so adjacent sides meet at 90°.
More questions like this
- (i) If P, Q, R are the midpoints of sides AB, AC, BC respectively of ∆ABC, show that ∆PQR is congruent to ∆QPA and to two other triangles which you should identify.
(ii) Suppose someone erases ∆ABC, leaving only ∆PQR on the paper. Can you reconstruct ∆ABC from ∆PQR? - In ∆ABC, let M and N be midpoints of AB and AC respectively. Let D be any point on BC. Show that MN bisects AD.
- In a quadrilateral ABCD, suppose AB ‖ DC and AB ≠ CD. Suppose G and H are the midpoints of AC and BD respectively. Prove that GH ‖ AB. (Why did we assume AB ≠ CD?)
- Suppose the midpoints of sides AB, BC, CD and DA of a quadrilateral ABCD are P, Q, R and S respectively.
- Suppose PQRS is the Varignon parallelogram of ABCD.
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