Question

consider right triangle $\triangle pqr$ below.

which expressions are equivalent to $\tan(\angle r)$?

choose 2 answers:

a $\dfrac{\text{length of side adjacent to }\angle q}{\text{length of side opposite to }\angle q}$

b $\dfrac{\text{length of side opposite to }\angle q}{\text{length of side adjacent to }\angle q}$

c $\dfrac{pq}{pr}$

d $\tan(\angle q)$

Explanation:

Step1: Define tangent for $\angle R$

In right $\triangle PQR$ (right at $P$), $\tan(\angle R) = \frac{\text{opposite to } \angle R}{\text{adjacent to } \angle R} = \frac{PQ}{PR}$

Step2: Analyze $\angle Q$ sides

For $\angle Q$, opposite side is $PR$, adjacent side is $PQ$. So $\frac{\text{adjacent to } \angle Q}{\text{opposite to } \angle Q} = \frac{PQ}{PR}$

Step3: Match with options

This matches option A and option C. Option B is $\tan(\angle Q)$, and option D is $\tan(\angle Q)$ which is $\frac{PR}{PQ}$, not equal to $\tan(\angle R)$.

Answer:

A. $\frac{\text{length of side adjacent to } \angle Q}{\text{length of side opposite to } \angle Q}$
C. $\frac{PQ}{PR}$