In the figure, DE ∥ BC in △ABC. Given AD = 1.7 cm, AB = 6.8 cm, AC = 9 cm. Find AE (in cm).
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A2.25cm
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B4.5cm
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C1.25cm
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D2.5cm
Answer
Correct Answer: 2.25cm
Explanation
Introduction / Context:When a segment DE is drawn parallel to BC in △ABC, triangles ADE and ABC are similar (Basic Proportionality Theorem/Thales). Corresponding sides are proportional.
Given Data / Assumptions:
- DE ∥ BC ⇒ △ADE ∼ △ABC.
- AD = 1.7, AB = 6.8, AC = 9.
- Need AE.
Concept / Approach:Use the similarity ratio AD/AB = AE/AC to compute AE directly.
Step-by-Step Solution:
AD/AB = AE/AC1.7 / 6.8 = AE / 91.7/6.8 = 0.25 ⇒ AE = 9 * 0.25 = 2.25 cmVerification / Alternative check:Scale factor from big to small is 0.25 (since AD is one-fourth of AB); then AE must be one-fourth of AC, i.e., 2.25 cm.
Why Other Options Are Wrong:They correspond to using wrong sides or inverting the ratio; only 2.25 cm respects the similarity scale factor consistently.
Common Pitfalls:Using BD/BC or mixing sides from different triangles; misreading AD/AB as AB/AD.
Final Answer:2.25cm