Question

timed problem
score:0/10
find: tan a
the figure is not drawn to scale
answer
\\(\frac{7}{24}\\)
\\(\frac{24}{25}\\)
\\(\frac{7}{25}\\)
\\(\frac{24}{7}\\)

Explanation:

Step1: Recall tangent definition

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

Step2: Identify sides for $\angle A$

For $\angle A$, opposite side is $24$, adjacent side is $7$.

Step3: Calculate $\tan A$

$\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{7}$? Wait, no—wait, wait. Wait, the right angle is at the corner with sides 7 and 24. So angle A: opposite side to A is 24, adjacent is 7? Wait, no, let's check the triangle. The right triangle has legs 7 and 24, hypotenuse 25. Angle A is at the vertex with leg 7 (adjacent) and leg 24 (opposite)? Wait, no: in the triangle, the right angle is between the sides of length 7 and 24. So angle A is at the end of the side of length 7. So for angle A, the opposite side is 24 (the side not adjacent to A), and the adjacent side is 7 (the side forming angle A with the hypotenuse). Wait, no, tangent is opposite over adjacent. So opposite to A is 24, adjacent is 7? Wait, no, wait: let's label the triangle. Right angle at, say, C. Then side AC = 7, BC = 24, AB = 25. Then angle A is at vertex A, between AC (7) and AB (25). So the opposite side to angle A is BC (24), adjacent side is AC (7). So $\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{7}$? Wait, but the options have $\frac{24}{7}$ as one of them? Wait, the options are $\frac{7}{24}$, $\frac{24}{25}$, $\frac{7}{25}$, $\frac{24}{7}$. Wait, no, wait—maybe I mixed up. Wait, no: angle A, the sides: adjacent is the leg next to A (not the hypotenuse), opposite is the leg opposite A. So if angle A is at the vertex with leg 7 (so that leg is adjacent), then the opposite leg is 24. So $\tan A = \frac{24}{7}$. Wait, but let's re-express. Wait, maybe I had it backwards. Wait, no: in a right triangle, for angle A, the two legs: one is adjacent (along with hypotenuse), one is opposite. So if angle A is at (let's say) point A, with AC = 7 (adjacent), BC = 24 (opposite), then $\tan A = \frac{BC}{AC} = \frac{24}{7}$. Yes. So $\tan A = \frac{24}{7}$. Wait, but the first option is $\frac{7}{24}$, second $\frac{24}{25}$, third $\frac{7}{25}$, fourth $\frac{24}{7}$. So the correct answer is $\frac{24}{7}$. Wait, but let me check again. Wait, maybe I flipped. Wait, no: adjacent is the leg that is part of angle A. So angle A is between the side of length 7 (adjacent) and the hypotenuse. The other leg (24) is opposite angle A. So tangent is opposite over adjacent, so 24/7. So the fourth option, $\frac{24}{7}$.

Answer:

$\frac{24}{7}$ (the fourth option, which is $\boldsymbol{\frac{24}{7}}$)