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High School Math : How to find the length of the side of a right triangle

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Example Question #141 : Plane Geometry

Points $\dpi{100} \small A$ $\displaystyle \dpi{100} \small A$ , $\dpi{100} \small B$ $\displaystyle \dpi{100} \small B$ , and $\dpi{100} \small C$ $\displaystyle \dpi{100} \small C$ are collinear (they lie along the same line). $\angle ACD = 90^{\circ}$ $\displaystyle \angle ACD = 90^{\circ}$ , $\angle CAD=30^{\circ}$ $\displaystyle \angle CAD=30^{\circ}$ , $\angle CBD=60^{\circ}$ $\displaystyle \angle CBD=60^{\circ}$ , $\overline{AD}=4$ $\displaystyle \overline{AD}=4$

Find the length of segment $\overline{BD}$ $\displaystyle \overline{BD}$ .

Possible Answers:

$\frac{4\sqrt{3}}{3}$ $\displaystyle \frac{4\sqrt{3}}{3}$

$\frac{\sqrt{3}}{2}$ $\displaystyle \frac{\sqrt{3}}{2}$

$\frac{2\sqrt{3}}{3}$ $\displaystyle \frac{2\sqrt{3}}{3}$

$2$ $\displaystyle 2$

$2\sqrt{3}$ $\displaystyle 2\sqrt{3}$

Correct answer:

$\frac{4\sqrt{3}}{3}$ $\displaystyle \frac{4\sqrt{3}}{3}$

Explanation:

The length of segment $\overline{BD}$ $\displaystyle \overline{BD}$ is $\frac{4\sqrt{3}}{3}$ $\displaystyle \frac{4\sqrt{3}}{3}$

Note that triangles $\dpi{100} \small ACD$ $\displaystyle \dpi{100} \small ACD$ and $\dpi{100} \small BCD$ $\displaystyle \dpi{100} \small BCD$ are both special, 30-60-90 right triangles. Looking specifically at triangle $\dpi{100} \small ACD$ $\displaystyle \dpi{100} \small ACD$ , because we know that segment $\overline{AD}$ $\displaystyle \overline{AD}$ has a length of 4, we can determine that the length of segment $\overline{CD}$ $\displaystyle \overline{CD}$ is 2 using what we know about special right triangles. Then, looking at triangle $\dpi{100} \small BCD$ $\displaystyle \dpi{100} \small BCD$ now, we can use the same rules to determine that segment $\overline{BD}$ $\displaystyle \overline{BD}$ has a length of $\frac{4}{\sqrt{3}}$ $\displaystyle \frac{4}{\sqrt{3}}$

which simplifies to $\frac{4\sqrt{3}}{3}$ $\displaystyle \frac{4\sqrt{3}}{3}$ .

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High School Math : How to find the length of the side of a right triangle

Study concepts, example questions & explanations for High School Math

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Example Question #141 : Plane Geometry

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