Question 6·Hard·Right Triangles and Trigonometry

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In the figure above, $\sin(\angle ACD)=\dfrac{5}{13}$ and the length of $\overline{CD}$ is 60.

Which choice is the length of $\overline{AB}$ ?

When a diagram shows a right triangle with an altitude drawn from the right angle to the hypotenuse, look for two moves: (1) use the given trig ratio to recognize a special right-triangle ratio (like $5$ - $12$ - $13$ ) in one of the smaller triangles to find missing side lengths, and (2) use similarity/altitude relationships such as $\text{(leg)}^2 = (\text{hypotenuse})(\text{adjacent hypotenuse segment})$ to connect the small triangle to the entire triangle.

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Identify the right triangle that uses the given sine

The sine expression involves $\angle ACD$ , so focus on triangle $ACD$ and use the right-angle marking at $D$ .

Use a known right-triangle ratio

If $\sin(\angle ACD)=\dfrac{5}{13}$ , then the sides of triangle $ACD$ follow a $5$ - $12$ - $13$ pattern (scaled by some factor). Use $CD=60$ to find that factor.

Connect triangle $ACD$ to triangle $ABC$

Because $CD$ is drawn to the hypotenuse, triangle $ACD$ is similar to the large right triangle. Use the relationship $AC^2=AB\cdot AD$ .

Desmos Guide

Compute the scale factor

In Desmos, type k=60/12 to compute the scale factor $k$ .

Compute $AD$ and $AC$ from the $5$ - $12$ - $13$ ratio

Type a_d=5k and a_c=13k (Desmos will treat these as variables $a_d$ and $a_c$ ).

Use the altitude relationship to compute $AB$

Type a_b=a_c^2/a_d and read the value of a_b.

Step-by-step Explanation

Use the sine ratio in right triangle $ACD$

From the diagram, $\angle ADC$ is a right angle, so triangle $ACD$ is a right triangle.

For $\angle ACD$ ,

\sin(\angle ACD)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{AD}{AC}=\frac{5}{13}

So triangle $ACD$ matches the $5$ - $12$ - $13$ ratio:

AD=5k,\quad CD=12k,\quad AC=13k

Use $CD=60$ to find the scale factor

Since $CD=12k$ and $CD=60$ ,

12k=60 \Rightarrow k=5

Then

AD=5k=25 \quad \text{and} \quad AC=13k=65

Relate a leg to the hypotenuse using the altitude

From the diagram, triangle $ABC$ is a right triangle with an altitude from $C$ to the hypotenuse $\overline{AB}$ at point $D$ . In this situation, triangle $ACD$ is similar to triangle $ABC$ , which leads to the geometric-mean relationship

AC^2 = AB\cdot AD

Solve for $AB$

Substitute $AC=65$ and $AD=25$ into $AC^2=AB\cdot AD$ :

65^2 = AB\cdot 25

AB = \frac{65^2}{25} = \frac{4225}{25} = 169

Therefore, the length of $\overline{AB}$ is 169.

Question 6·Hard·Right Triangles and Trigonometry

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation