AMC 8 2005 Problem 9

AMC8 2005 年第 9 题

三角形★★☆☆☆
In quadrilateral ABCD , sides AB‾\overline{AB} and BC‾\overline{BC} both have length 10, sides CD‾\overline{CD} and DA‾\overline{DA} both have length 17, and the measure of angle ADC is 60∘60^\circ. What is the length of diagonal AC‾\overline{AC}?
四边形 ABCDABCD 中，边 AB‾\overline{AB} 和 BC‾\overline{BC} 的长度都是 10，边 CD‾\overline{CD} 和 DA‾\overline{DA} 的长度都是 17，且 ∠ADC=60∘\angle ADC = 60^\circ。那么对角线 AC‾\overline{AC} 的长度是多少？
AMC 8 2005 Problem 9 figure
  1. A.13.5
  2. B.14
  3. C.15.5
  4. D.17
  5. E.18.5

正确答案 · Correct Answer

D

解析 · Solution

Solution 1

Because AD‾=CD‾\overline{AD} = \overline{CD}, △ADC\triangle ADC is an isosceles triangle with ∠DAC=∠DCA\angle DAC = \angle DCA. Angles in a triangle add up to 180∘180^\circ, and since ∠ADC=60∘\angle ADC=60^\circ, the other two angles are also 60∘60^\circ, and △ADC\triangle ADC is an equilateral triangle. Therefore AC‾=DA‾=17\overline{AC}=\overline{DA}=17.

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