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Congruent and Similar Triangles

If you want to find similar triangles, use only SSS, SAS and AAA. Don't make an ASS of yourself.

Problem Solving

         

Let \(\lvert\overline {AB}\rvert\) denote the length of \(\overline {AB}.\) Then in the above diagram, \(\lvert\overline {AB}\rvert = \lvert\overline {BC}\rvert\) and \(\lvert\overline {BF}\rvert = \lvert\overline {FE}\rvert.\) If \(\lvert\overline{CD}\rvert=38,\) what is \(\lvert\overline {DE}\rvert?\)

In the above diagram, \(\angle ABD = \angle BCE = \angle CAF.\) Given the lengths \[\lvert \overline{AB}\rvert=12, \lvert \overline{AC}\rvert =7, \lvert \overline{BC}\rvert = 14,\] what is \(\lvert \overline{DE}\rvert : \lvert \overline{EF}\rvert : \lvert \overline{DF}\rvert?\)

Note: The above diagram is not drawn to scale.

In the above quadrilateral \(\square ABCD,\) \[\overline{AD} \parallel \overline{EF} \parallel \overline{BC}, \lvert\overline{AE}\rvert = 2\lvert\overline{EB}\rvert, \lvert\overline{BM}\rvert = 4, \lvert\overline{AD}\rvert = 6, \lvert\overline{BC}\rvert = 9,\] where \(\lvert\overline{AE}\rvert\) denotes the length of \(\overline{AE}.\) What is \(\lvert\overline{DO}\rvert?\)

Note: The above diagram is not drawn to scale.

In the above diagram, \(\triangle ABC\) is a right-angled triangle where \(\angle C\) is a right angle and \(\lvert{\overline{AC}}\rvert=\lvert{\overline{BC}}\rvert.\) If \(\lvert{\overline{AC}}\rvert=\lvert{\overline{AD}}\rvert,\) \(\overline{AB} \perp \overline{DE}\) and \(\lvert{\overline{CE}}\rvert=7,\) what is the measure of \(\overline{DB} ?\)

\(\triangle ABC\) is a right triangle with side lengths \[\lvert\overline{AB}\rvert = 25, \lvert\overline{BC}\rvert = 50\] If \(\square{DBEF}\) in the above diagram is a square, what is the area of \(\square{DBEF}?\)

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