Question

find: \\( \tan b \\) \\( \text{the figure is not drawn to scale.} \\) answer \\( \frac{14}{50} \\) \\( \frac{48}{50} \\) \\( \frac{14}{48} \\) \\( \frac{48}{14} \\)

Explanation:

Step1: Recall tangent definition

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

Step2: Identify sides for $\angle B$

For $\angle B$, opposite side is $48$, adjacent side is $14$.

Step3: Calculate $\tan B$

$\tan B = \frac{\text{opposite to } B}{\text{adjacent to } B} = \frac{48}{14}$? Wait, no—wait, let's recheck. Wait, the right triangle: right angle, so sides: one leg is 14 (horizontal), one leg is 48 (vertical), hypotenuse 50. For angle B: the angle at the top (B), so opposite side to B is the vertical leg (48)? Wait no, wait: angle B is at the top vertex. So the sides: adjacent to B is the horizontal leg (14), opposite is the vertical leg (48)? Wait no, wait: in the triangle, angle B: the sides: the leg adjacent to B is the one forming angle B with the hypotenuse, and opposite is the other leg. Wait, let's label the triangle: right angle at the left, so vertices: left (right angle), bottom (A), top (B). So side from left to bottom (A) is 48 (vertical), left to top (B) is 14 (horizontal), hypotenuse from A to B is 50. So angle at B: the sides: adjacent to B is the horizontal side (14), opposite is the vertical side (48). Wait, no: tangent of angle B is opposite over adjacent. So opposite to B is the side opposite angle B, which is the side from left to A (length 48), and adjacent is the side from left to B (length 14). So $\tan B = \frac{48}{14}$? Wait, but let's check the options. Wait the options include $\frac{48}{14}$? Wait the last option is $\frac{48}{14}$. Wait, but let's re-express: in a right triangle, for angle B, the opposite side is the leg not connected to B, so if B is at the top, the opposite side is the vertical leg (48), adjacent is the horizontal leg (14). So $\tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{48}{14}$. Wait, but let's confirm:

Wait, the triangle: right-angled at the left. So vertices: L (right angle), A (bottom), B (top). So sides: LA = 48 (vertical), LB = 14 (horizontal), AB = 50 (hypotenuse). Angle at B: between LB (14) and AB (50). So the side opposite angle B is LA (48), adjacent is LB (14). So $\tan B = \frac{LA}{LB} = \frac{48}{14}$. So that's the fourth option? Wait the options are:

1. $\frac{14}{50}$

1. $\frac{48}{50}$

1. $\frac{14}{48}$

1. $\frac{48}{14}$

So the correct answer is the fourth option, $\frac{48}{14}$. Wait, but let's check again. Wait, maybe I mixed up adjacent and opposite. Wait, angle B: the sides: the leg adjacent to B is the one that is part of angle B (other than hypotenuse), so the leg from B to L (length 14) is adjacent, and the leg from L to A (length 48) is opposite. So yes, $\tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{48}{14}$. So the correct option is the last one, $\frac{48}{14}$.

Answer:

$\frac{48}{14}$ (the last option, i.e., the one with $\frac{48}{14}$)