Question

timed problem score: 2/10 current time: 208.3 find: tan b the figure is not drawn to scale. right triangle with legs 24, 45, hypotenuse 51, right angle, angle b answer options: \\(\frac{24}{51}\\), \\(\frac{45}{24}\\), \\(\frac{24}{45}\\), \\(\frac{45}{51}\\)

Explanation:

Step1: Recall tangent definition

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

Step2: Identify sides for angle B

For angle \( B \), opposite side is \( 45 \), adjacent side is \( 24 \).

Step3: Calculate $\tan B$

$\tan B = \frac{\text{opposite to } B}{\text{adjacent to } B} = \frac{45}{24}$ (Wait, no, wait: Wait, the right triangle has legs 24 and 45, hypotenuse 51. Wait, angle B: let's check the triangle. The right angle is between 24 and 45. So angle B: the side opposite to B is 45? Wait no, wait the triangle: let's label. The right angle is at the vertex with sides 24 and 45. So angle B: the side opposite to B is 45? Wait no, wait, the side adjacent to B: let's see. Wait, in the right triangle, angle B: the sides: the leg opposite to B is 45? Wait no, wait the leg adjacent to B is 24? Wait no, wait, let's re-express. Wait, the triangle has sides: one leg is 24 (vertical), one leg is 45 (horizontal), hypotenuse 51. So angle B is at the top, so the side opposite to angle B is the horizontal leg (45), and the side adjacent to angle B is the vertical leg (24). Wait, no: tangent of angle B is opposite over adjacent. So opposite to B is 45, adjacent is 24? Wait no, wait, no: wait, in a right triangle, for angle B, the opposite side is the side not connected to B, and adjacent is the side connected to B (other than hypotenuse). Wait, the right angle is between 24 and 45. So angle B is at the end of the hypotenuse and the side of length 24. So the side opposite to angle B is 45 (the horizontal leg), and the side adjacent to angle B is 24 (the vertical leg). Wait, but wait, no: let's use SOHCAHTOA. Tangent is opposite over adjacent. So for angle B: opposite side is 45, adjacent side is 24? Wait, but wait, the options include 45/24, 24/45, etc. Wait, maybe I made a mistake. Wait, no: wait, the triangle: the right angle is at the bottom left, so angle A is at bottom left, angle B is at top, hypotenuse is 51. So side opposite angle B is 45 (the horizontal side from right angle to angle A), and side adjacent to angle B is 24 (the vertical side from right angle to angle B). So $\tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{45}{24}$? Wait, but let's check the options. The options are 24/51, 45/24, 24/45, 45/51. Wait, no, wait the options given are $\frac{24}{51}$, $\frac{45}{24}$, $\frac{24}{45}$, $\frac{45}{51}$. Wait, maybe I mixed up opposite and adjacent. Wait, angle B: the sides: the leg opposite to B is 45? No, wait, the leg adjacent to B is 45? Wait, no, let's draw the triangle mentally. Right angle at C (let's say), so sides: AC = 45, BC = 24, AB = 51. Then angle B is at vertex B. So in triangle ABC, right-angled at C. Then angle B: the opposite side is AC (45), adjacent side is BC (24). So $\tan B = \frac{AC}{BC} = \frac{45}{24}$. Wait, but let's check the options. One of the options is $\frac{45}{24}$. Wait, but let's simplify? No, the options are as given. Wait, but maybe I made a mistake. Wait, no, tangent is opposite over adjacent. So for angle B, opposite is 45, adjacent is 24, so $\tan B = \frac{45}{24}$. Wait, but let's check the options. The options are:

1. $\frac{24}{51}$

2. $\frac{45}{24}$

3. $\frac{24}{45}$

4. $\frac{45}{51}$

Wait, so the correct one should be $\frac{45}{24}$? Wait, no, wait, maybe I got the angle wrong. Wait, angle B: maybe the opposite side is 24 and adjacent is 45? Wait, that would be $\frac{24}{45}$. Wait, let's re-examine. If angle B is at the top, then the side opposite to B is the horizontal side (45) and adjacent is vertical (24). Wait, no, maybe the triangle is labeled differently. Wait, the problem says "the figure is not drawn to scale". The triangle has a right angle, one leg is 24, one leg is 45, hypotenuse 51. So in a right triangle, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$. So for angle B, let's see: the sides: the leg opposite to B is 45 (since it's opposite the angle at B), and the leg adjacent to B is 24 (since it's adjacent to angle B, forming the angle with the hypotenuse). Wait, but maybe I have it reversed. Wait, let's take angle A: angle A is at the bottom, so opposite side is 24, adjacent is 45, so $\tan A = \frac{24}{45}$. Then angle B: opposite side is 45, adjacent is 24, so $\tan B = \frac{45}{24}$. So the correct answer should be $\frac{45}{24}$.

Answer:

$\frac{45}{24}$ (or simplified as $\frac{15}{8}$, but among the options, $\frac{45}{24}$ is present)