The altitude from the right angle of a right triangle to its hypotenuse splits the hypotenuse into two segments. There is a special relationship between the altitude and those two segments—can you write an equation involving $QS$, $PS$, and $SR$?

Once you have the length of $RS$, use the definition of tangent in right triangle $RQS$: $\tan(\text{angle at }Q)$ is a ratio of two sides. Which sides of $\triangle RQS$ should you use?

In an expression line, type a = 12^2/9. The value of a is the length of $RS$, coming from the equation $12^2 = 9 \cdot RS$.

In a new line, type a/12 to represent $\dfrac{RS}{QS}$. The value shown is $\tan \angle RQS$; if you tap it, you can convert it to a fraction and match it to one of the answer choices.

Draw right triangle $PQR$ with $\angle Q = 90^\circ$ and hypotenuse $PR$. Point $S$ lies on $PR$ so that $QS$ is an altitude, so $QS \perp PR$.

Focus on the smaller right triangle $RQS$:

• It is right at $S$.
• The angle we care about is $\angle RQS$ (at $Q$).
• In $\triangle RQS$:
  • The hypotenuse is $RQ$.
  • The side adjacent to $\angle RQS$ (but not the hypotenuse) is $QS = 12$.
  • The side opposite $\angle RQS$ is $RS$ (unknown).

In a right triangle, the altitude from the right angle to the hypotenuse has this property:

Here $QS = 12$ and $PS = 9$, so:

Compute $12^2$ and solve this equation for $SR$ to get the length of $RS$. Keep this value for the next step.

In right triangle $RQS$, for angle $\angle RQS$ at $Q$:

• Opposite side is $RS$.
• Adjacent side (non-hypotenuse) is $QS = 12$.

So,

Substitute $RS$ from Step 2 and $QS = 12$:

So the correct answer is $\dfrac{4}{3}$.