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

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Example Question #141 : Act Math

Triangles

Points \dpi{100} \small A, \dpi{100} \small B, and \dpi{100} \small C are collinear (they lie along the same line). , ,

Find the length of segment \overline{BD}.

Possible Answers:

\frac{2\sqrt{3}}{3}

2\sqrt{3}

2

\frac{\sqrt{3}}{2}

\frac{4\sqrt{3}}{3}

Correct answer:

\frac{4\sqrt{3}}{3}

Explanation:

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

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

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

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