Question

find the tangent of $\angle x$. simplify your answer and write it as a proper fraction, improper fraction, or whole number. $\tan(x) = $

Explanation:

Step1: Recall tangent definition

In a right triangle, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ to $\theta$.

Step2: Identify sides for $\angle X$

For $\angle X$, opposite side is $WV = 35$, adjacent side is $XW = 12$.

Step3: Calculate $\tan(X)$

$\tan(X) = \frac{\text{opposite}}{\text{adjacent}} = \frac{35}{12}$? Wait, no, wait. Wait, $XW$ is 12 (vertical), $WV$ is 35 (horizontal). Wait, $\angle X$: right angle at $W$, so triangle $XWV$ right-angled at $W$. So $\angle X$: sides: $XW = 12$ (adjacent), $WV = 35$ (opposite), $XV = 37$ (hypotenuse). So $\tan(X) = \frac{\text{opposite to } X}{\text{adjacent to } X} = \frac{WV}{XW} = \frac{35}{12}$? Wait, no, wait, $XW$ is 12 (length from $X$ to $W$), $WV$ is 35 (length from $W$ to $V$). So for angle $X$, the opposite side is $WV$ (35), adjacent is $XW$ (12). So $\tan(X) = \frac{35}{12}$? Wait, but let's check: in right triangle, tangent of angle is opposite over adjacent. So angle at $X$: the sides: opposite is $WV$ (35), adjacent is $XW$ (12). So yes, $\tan(X) = 35/12$? Wait, but 12-35-37 is a Pythagorean triple: $12^2 + 35^2 = 144 + 1225 = 1369 = 37^2$, correct. So $\tan(X) = \frac{35}{12}$. Wait, but wait, maybe I mixed up. Wait, $XW$ is 12 (vertical), $WV$ is 35 (horizontal). So angle at $X$: the sides: adjacent is $XW$ (12), opposite is $WV$ (35). So yes, $\tan(X) = 35/12$. Wait, but let me confirm the definition again. Tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. So for angle $X$, opposite side is $WV$ (35), adjacent is $XW$ (12). So $\tan(X) = 35/12$. Wait, but 35 and 12 are coprime? 35 factors: 5,7; 12: 2,2,3. So yes, 35/12 is the tangent. Wait, but wait, maybe I made a mistake. Wait, $XW$ is 12, $WV$ is 35. So angle at $X$: the triangle has vertices $X$, $W$, $V$, right-angled at $W$. So sides: $XW = 12$, $WV = 35$, $XV = 37$. So angle at $X$: between $XW$ and $XV$. So the opposite side to angle $X$ is $WV$ (35), adjacent is $XW$ (12). So $\tan(X) = 35/12$. Yes.

Answer:

$\frac{35}{12}$