ΔABC ~ ΔPQR, with AB = 6, BC = 8, and PQ = 9. A student sets up the proportion 6/9 = 8/QR to find QR. What is the student doing correctly or incorrectly?
AThis proportion is incorrect — the student should use AB/BC = PQ/QR (ratios within each triangle)
BThis proportion is correct — AB corresponds to PQ and BC corresponds to QR, giving QR = 12
CThis proportion is incorrect — the student should use AB/PQ = BC/QR but with the triangles switched
DThis proportion cannot be solved without knowing a third side
In ΔABC ~ ΔPQR, A↔P, B↔Q, C↔R, so AB corresponds to PQ and BC corresponds to QR. The proportion AB/PQ = BC/QR gives 6/9 = 8/QR, so QR = 12. This is the correct cross-triangle proportion matching corresponding sides. Option A describes ratios within a single triangle, which also works but is a different setup.
Question 2 Multiple Choice
In triangle ABC, segment DE is parallel to BC with D on AB and E on AC. If AD = 4, DB = 6, and AE = 5, what is EC?
AEC = 3
BEC = 7.5
CEC = 5
DEC = 4
By the Side-Splitter Theorem, a line parallel to one side of a triangle divides the other two sides proportionally: AD/DB = AE/EC. So 4/6 = 5/EC, giving EC = 5 × 6/4 = 7.5. The parallel line creates the same ratio on both sides of the triangle.
Question 3 True / False
If ΔABC ~ ΔDEF and AB/DE = BC/EF, it is possible that AC/DF is a different ratio.
TTrue
FFalse
Answer: False
When two triangles are similar, ALL pairs of corresponding sides share the same scale factor k. If AB/DE = BC/EF = k, then AC/DF must also equal k. The scale factor is a single constant that relates every pair of corresponding sides simultaneously. Two of the three ratios being equal forces the third to be equal as well — this is a direct consequence of similarity.
Question 4 True / False
When setting up proportions for similar triangles, you should identify corresponding sides using the similarity statement's vertex correspondence rather than matching sides that look similar in position in the diagram.
TTrue
FFalse
Answer: True
Diagrams can be drawn in many orientations — a side that appears 'on the left' in one triangle may correspond to the side 'on the right' in the other. The similarity statement (e.g., ΔABC ~ ΔDEF) encodes the exact correspondence: A↔D, B↔E, C↔F, giving AB↔DE, BC↔EF, AC↔DF. Using the statement prevents mismatches that arise from reading visual position alone.
Question 5 Short Answer
Explain why all three pairs of corresponding sides must share the same scale factor when two triangles are similar, rather than just requiring two pairs to match.
Think about your answer, then reveal below.
Model answer: Similarity means one triangle is a uniform scaling of the other — every distance is multiplied by the same scale factor k. This is not a property that can hold for two sides without holding for the third, because all three sides are transformed by the same multiplicative factor simultaneously. If two pairs of sides have the same ratio but the third does not, the triangles have different shapes — they are not similar.
In practice, once you establish similarity (via AA, SSS similarity, or SAS similarity), you can use any one pair of corresponding sides to find k, then apply that k to all other pairs. The three equal ratios are not independent facts but three expressions of a single underlying transformation — the scaling that maps one triangle onto the other.