Question 167·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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In right triangle $\triangle ABC$ , the measure of angle $B$ is $90^\circ$ , and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$ .

If $BD=6$ and $BC=10$ , which choice is the value of $\tan A$ ?

An altitude from a right angle to the hypotenuse creates similar right triangles. Match the angle you need to an angle in a smaller triangle, then identify its opposite and adjacent legs. If the known lengths fit a Pythagorean triple, use it to find the missing leg. Don't treat the altitude as a side of the original triangle.

Hints

Hint 1
The altitude makes a right angle at $D$ . The large triangle and $\triangle BCD$ also share angle $C$ , so they're similar: which angle in the smaller triangle matches angle $A$ ?
Hint 2
In $\triangle BCD$ , $BC$ is the hypotenuse because it faces the right angle at $D$ . The lengths $6$ and $10$ fit a doubled $3$ - $4$ - $5$ right triangle. What's the other leg?
Hint 3
For the angle in the smaller triangle that matches $A$ , tangent means opposite leg over adjacent leg. Look at $BD$ and $CD$ : which one lies across from that angle?

Step-by-step

Match the angle in the smaller triangle
Step 1
Identify the similar triangles
No Desmos needed. The known lengths fit a scaled $3$ - $4$ - $5$ right triangle.

The altitude $BD$ meets $AC$ at a right angle. The large triangle and $\triangle BCD$ share angle $C$ , and each has a right angle. So they're similar, meaning their angles match: $\triangle ABC\sim\triangle BDC$ .
Step 2
Find the angle that matches $A$
In the matching order $\triangle ABC\sim\triangle BDC$ , the right angle $B$ matches $D$ , and $C$ matches $C$ . So angle $A$ matches $\angle CBD$ . You can find $\tan A$ using the sides of that smaller triangle.
Step 3
Find the missing leg
In $\triangle BCD$ , $BC=10$ is the hypotenuse, across from the right angle at $D$ , and $BD=6$ is a leg. These fit the $6$ - $8$ - $10$ triple, which is twice $3$ - $4$ - $5$ . So $CD=8$ . Don't subtract $10-6$ to find a leg; right-triangle side lengths don't work that way.
Step 4
Take opposite over adjacent
Tangent is the opposite leg divided by the adjacent leg. From $\angle CBD$ , $CD$ is opposite and $BD$ is adjacent, so
$\tan A=\tan\angle CBD=\frac{CD}{BD}=\frac{8}{6}=\frac{4}{3}.$
The value of $\tan A$ is $\frac{4}{3}$ . Choice B.

Question 167·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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