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Geometry

# Triangles - Identify Similar Triangles

In the above diagram, if $$a=18,$$ $$b=12,$$ $$c=14$$ and $$d=8,$$ what is the measure of $$\overline{AD} ?$$

Note: The above diagram is not drawn to scale.

Given $$\triangle PQR$$ with side lengths $$\lvert{\overline{PQ}}\rvert=16\text{ cm},$$ $$\lvert{\overline{QR}}\rvert=20\text{ cm}$$ and $$\angle PQR=60^\circ,$$ which of the following triangles is similar to $$\triangle PQR ?$$

(a) $$\triangle ABC$$ with side lengths $$\lvert{\overline{AB}}\rvert={15}\text{ cm},$$ $$\lvert{\overline{AC}}\rvert={12}\text{ cm}$$ and $$\angle BAC=60^\circ$$
(b) $$\triangle DEF$$ with side lengths $$\lvert{\overline{DE}}\rvert={8}\text{ cm},$$ $$\lvert{\overline{DF}}\rvert={10}\text{ cm}$$ and $$\angle EDF=80^\circ$$
(c) $$\triangle GHI$$ with angles $$\angle GHI=60^\circ$$ and $$\angle GIH=80^\circ$$
(d) $$\triangle JKL$$ with side lengths $$\lvert{\overline{JK}}\rvert={20}\text{ cm},$$ $$\lvert{\overline{KL}}\rvert={18}\text{ cm}$$ and $$\lvert{\overline{JL}}\rvert={16}\text{ cm}$$

In the above diagram, $$x=21,$$ $$y=14,$$ $$\alpha={35}^\circ,$$ and $$\beta={75}^\circ.$$ Which condition would mean that the two triangles are similar?

Note: The above diagram is not drawn to scale.

In the above diagram, if $$a=30,$$ $$b=25$$ and $$\angle BAC=\angle AHC=90^\circ,$$ what is the measure of $$\overline{HC} ?$$

Note: The above diagram is not drawn to scale.

In the above diagram, if $$a=45,$$ $$b=30$$ and $$\angle ABC=\angle DAC,$$ what is the measure of $$\overline{BD} ?$$

Note: The above diagram is not drawn to scale.

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