Question

in the diagram below, \\(\overline{ef}\\) is parallel to \\(\overline{bc}\\). if \\(df\\) is 15 less than \\(ed\\), \\(bd = 63\\), and \\(dc = 36\\), find the length of \\(\overline{ed}\\). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: 识别相似三角形

因为\(\overline{EF} \parallel \overline{BC}\)，且\(\angle C = \angle F = 90^\circ\)，\(\angle D\)公共，所以\(\triangle DEF \sim \triangle DBC\)（AA相似）。
根据相似三角形的性质，对应边成比例，即\(\frac{ED}{BD}=\frac{FD}{CD}\)。

Step2: 设未知数并代入

设\(ED = x\)，则\(FD = x - 15\)（因为\(DF\)比\(ED\)小\(15\)）。已知\(BD = 63\)，\(CD = 36\)，代入比例式得：
\(\frac{x}{63}=\frac{x - 15}{36}\)

Step3: 解比例方程

交叉相乘得：\(36x = 63(x - 15)\)
展开右边：\(36x = 63x - 945\)
移项：\(63x - 36x = 945\)
合并同类项：\(27x = 945\)
两边同时除以\(27\)：\(x=\frac{945}{27}=35\)

Answer:

\(35\)