(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?
- (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?
Step-by-step solution
Idea: The Midpoint Theorem makes each side of the middle triangle equal to half a side of ∆ABC, which is also the length of the half-sides at the corners. Then SSS matches the triangles. For (ii), each side of ∆ABC passes through a vertex of ∆PQR and is parallel to the opposite side of ∆PQR.
(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.
- Midpoint Theorem: in ∆ABC, PQ = ½BC = BR = RC; QR = ½AB = AP = PB; RP = ½AC = AQ = QC.1 mark
- ∆PQR and ∆QPA: PQ = QP (common), QR = PA, RP = AQ. So ∆PQR ≅ ∆QPA (SSS).1 mark
- Likewise ∆PQR ≅ ∆RBP (PQ = RB, QR = BP, RP = PR) and ∆PQR ≅ ∆CRQ (PQ = CR, QR = RQ, RP = QC), both by SSS.½ mark
(ii) Suppose someone erases ∆ABC, leaving only ∆PQR on the paper. Can you reconstruct ∆ABC from ∆PQR?
- By the Midpoint Theorem, AB ‖ QR and AB passes through P. There is only one line through P parallel to QR, so line AB is the line through P parallel to QR. Similarly AC is the line through Q parallel to PR, and BC is the line through R parallel to PQ.1 mark
- So: draw the three lines through P, Q, R parallel to QR, PR, PQ respectively. Their crossing points are A, B, C. Yes, ∆ABC can be reconstructed.½ mark
Answer to write in the exam
(i)
PQ = ½BC = BR = RC, QR = ½AB = AP = PB, RP = ½AC = AQ = QC (Midpoint Theorem)
∆PQR ≅ ∆QPA (SSS: PQ = QP, QR = PA, RP = AQ)
∆PQR ≅ ∆RBP (SSS: PQ = RB, QR = BP, RP = PR)
∴ ∆PQR ≅ ∆CRQ (SSS: PQ = CR, QR = RQ, RP = QC)
(ii)
QR ‖ AB, PR ‖ AC, PQ ‖ BC (Midpoint Theorem)
Line AB = line through P ‖ QR; AC = line through Q ‖ PR; BC = line through R ‖ PQ (unique parallel through a point)
∴ Yes: draw these three lines; they meet at A, B, C.
Common mistakes that cost marks
- Writing the congruence with the wrong vertex order, e.g. ∆PQR ≅ ∆APQ. The order must match equal sides: P↔Q, Q↔P, R↔A.
- Saying the four triangles are congruent “because they look the same”. Each pair needs three equal sides from the Midpoint Theorem.
- In (ii), joining P, Q, R to points chosen by eye. The reconstruction must use the parallel lines.
How this can come in the exam
P, Q, R are the midpoints of sides AB, AC, BC of ∆ABC, with AB = 8 cm, BC = 10 cm, CA = 12 cm. The side QR equals
- 4 cm
- 5 cm
- 6 cm
- 8 cm
Show answer
(A) 4 cm
QR joins the midpoints of AC and BC, so QR = ½AB = 4 cm.
Try one yourself
The triangle formed by joining the midpoints of the sides of ∆XYZ has sides 3 cm, 4 cm and 5 cm. Find the sides of ∆XYZ.
Show answer
Each side of ∆XYZ is twice a side of the small triangle: 6 cm, 8 cm and 10 cm.
More questions like this
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