Try to prove directly that the four smaller triangles are congruent. You will see that no congruence test applies, because many quantities are unknown.
Answer: Comparing, say, ∆APQ and ∆PBR, we know only one pair of equal parts (AP = PB). We do not know AQ = PR, PQ = BR, or any equal angles, so SSS, SAS, ASA and RHS all fail. The missing facts (PQ = ½BC, QR = ½AB, RP = ½CA) come from the Midpoint Theorem, and then SSS works.
Step-by-step solution
Idea: A congruence test needs three matching parts. At the start we only know that the midpoints halve the sides. The segments PQ, QR, RP are new, and nothing yet tells us their lengths or directions.
- Let P, Q, R be the midpoints of AB, AC, BC. Try ∆APQ and ∆PBR. Known: AP = PB. Unknown: is AQ equal to PR? Is PQ equal to BR? Is ∠A equal to ∠BPR? Nothing given answers these.1 mark
- With only one pair of equal sides, none of SSS, SAS, ASA, AAS or RHS can be used. The same happens for every other pair of small triangles. So the direct attempt gets stuck.1 mark
- The way out is to first study one new segment at a time. The Midpoint Theorem shows PQ = ½BC = BR = RC, QR = ½AB = AP = PB and PR = ½AC = AQ = QC. Then each small triangle has sides ½AB, ½BC, ½CA, and all four are congruent by SSS.1 mark
A direct proof fails because only one pair of equal parts is known. Once the Midpoint Theorem gives PQ = ½BC, QR = ½AB and RP = ½CA, all four triangles are congruent by SSS.
Answer to write in the exam
∆APQ and ∆PBR: only AP = PB is known
AQ, PR, PQ, BR and the angles at P are unknown ⇒ no congruence test applies
Midpoint Theorem: PQ = ½BC, QR = ½AB, RP = ½CA
∴ Each small triangle has sides ½AB, ½BC, ½CA ⇒ all four congruent (SSS).
Common mistakes that cost marks
- Claiming ∆APQ ≅ ∆PBR by SAS using AP = PB and “the angle at A equals the angle at B”. Those angles are not equal in general.
- Assuming PQ = BR because they “look equal”. That is exactly what needs proof.
- Using the Midpoint Theorem to prove itself; here it is used only after it has been proved separately.
How this can come in the exam
Short answer (2 marks)
P, Q, R are the midpoints of sides AB, AC, BC of ∆ABC. Using the Midpoint Theorem, prove that ∆APQ ≅ ∆QRC.
Show answer
AQ = QC (Q is the midpoint). By the Midpoint Theorem, PQ = ½BC = RC and QR = ½AB = AP (1 mark). So ∆APQ ≅ ∆QRC by SSS (AP = QR, PQ = RC, AQ = QC) (1 mark).Try one yourself
In ∆ABC, AB = 10 cm, BC = 14 cm, CA = 12 cm. P, Q, R are midpoints of AB, AC, BC. Find the sides of ∆PBR.
Show answer
PB = 5 cm, BR = 7 cm, PR = ½AC = 6 cm.
More questions like this
- The segment joining the midpoints of two sides of a triangle is parallel to the third side and has half the length.
- Read the statement of the Midpoint Theorem again. The following questions seem natural to ask. What can we say about (1) a segment PQ (with P on side AB and Q on side AC) that is parallel to the third side and has half the length, and (2) the line that passes through the midpoint of one side and is parallel to another side? For now, experiment and see if you can make a guess about question (2).
- The line drawn through the midpoint of one side of a triangle and parallel to another side bisects the third side (and the resulting segment has half the length of the parallel side).
- BCRP is a parallelogram, so CR = BP. BP = PA and ∠AQP = ∠CQR (vertically opposite angles). Combining with CR ‖ BA, ∆APQ ≅ ∆CRQ. (Why?)
- We will later generalise the converse of the Midpoint Theorem by dropping the condition that P is the midpoint of AB and assuming only that PQ ‖ BC. Can you guess the conclusion in this case? (Hint: the name of the more general theorem includes the word proportionality.)
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