Geometry

Congruent and Similar Triangles

Similar Triangles Problem Solving

         

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

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

Note: The above diagram is not drawn to scale.

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In the above quadrilateral ABCD,ABCD, AD‾∥EF‾∥BC‾,\overline{AD} \parallel \overline{EF} \parallel \overline{BC}, and

∣AE‾∣=2∣EB‾∣,∣BM‾∣=4,∣AD‾∣=6,∣BC‾∣=9, \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 ∣AE‾∣\lvert\overline{AE}\rvert denotes the length of AE‾.\overline{AE}. What is ∣DO‾∣?\lvert\overline{DO}\rvert?

Note: The above diagram is not drawn to scale.

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

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△ABC\triangle ABC is a right triangle with side lengths ∣AB‾∣=25,∣BC‾∣=50\lvert\overline{AB}\rvert = 25, \lvert\overline{BC}\rvert = 50 If □DBEF\square{DBEF} in the above diagram is a square, what is the area of □DBEF?\square{DBEF}?

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