HMMT 二月 2013 · 几何 · 第 6 题
HMMT February 2013 — Geometry — Problem 6
题目详情
英文原题
Let ABCD be a quadrilateral such that ∠ABC =∠CDA = 90◦, and BC = 7. Let EandFbe on
BD such that AEandCF are perpendicular to BD. Suppose that BE = 3. Determine the product
of the smallest and largest possible lengths of DF.
解析
英文解析
- Let ABCD be a quadrilateral such that ∠ ABC = ∠ CDA = 90 , and BC = 7. Let E and F be on
BD such that AE and CF are perpendicular to BD . Suppose that BE = 3. Determine the productof the smallest and largest possible lengths of DF .
Answer: 9 By inscribed angles, ∠ CDB = ∠ CAB , and ∠ ABD = ∠ ACD . By definition,
∠ AEB = ∠ CDA = ∠ ABC = ∠ CF A . Thus, △ ABE ∼ △ ADC and △ CDF ∼ △ CAB . This showsthat
BE CD DF AB = and =
AB CA CD BD
Based on the previous two equations, it is sufficient to conclude that 3 = EB = F D . Thus, F D mustequal to 3, and the product of its largest and smallest length is 9.
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