Question

find: tan a
the figure is not drawn to scale.
answer
\frac{54}{72}
\frac{72}{90}
\frac{54}{90}
\frac{72}{54}

Explanation:

Step1: Recall tangent definition

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

Step2: Identify sides for $\angle A$

For $\angle A$, opposite side is $72$, adjacent side is $54$.

Step3: Calculate $\tan A$

$\tan A = \frac{72}{54}$? Wait, no—wait, opposite to $A$: wait, the right angle is between $54$ and $72$. So angle $A$: adjacent is $54$, opposite is $72$? Wait, no, let's check the triangle. The right triangle has legs $54$ (adjacent to $A$) and $72$ (opposite to $A$), hypotenuse $90$. So $\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{72}{54}$? Wait, no, wait: adjacent to $A$ is $54$, opposite is $72$? Wait, no, maybe I mixed up. Wait, angle $A$: the sides: the leg next to $A$ is $54$ (adjacent), the leg opposite is $72$. So $\tan A = \frac{72}{54}$? But wait, the options: one of the options is $\frac{72}{54}$? Wait, no, the options are $\frac{54}{72}$, $\frac{72}{90}$, $\frac{54}{90}$, $\frac{72}{54}$. Wait, let's re-express. Wait, $\tan A = \frac{\text{opposite}}{\text{adjacent}}$. Opposite to $A$: the side not connected to $A$ except hypotenuse. So in the right triangle, angle $A$: adjacent side is $54$ (the leg forming angle $A$ with hypotenuse), opposite side is $72$ (the other leg). So $\tan A = \frac{72}{54} = \frac{4}{3}$? But the options: wait, the options are $\frac{54}{72}$, $\frac{72}{90}$, $\frac{54}{90}$, $\frac{72}{54}$. Wait, maybe I flipped. Wait, no—wait, maybe adjacent is $72$ and opposite is $54$? No, that can't be. Wait, let's label the triangle: right angle at the top, so vertices: right angle (let's say $C$), $A$ at the bottom left, $B$ at the bottom right. So side $AC = 54$, $BC = 72$, $AB = 90$ (hypotenuse). Then angle $A$: between $AC$ (54) and $AB$ (90). So opposite side to $A$ is $BC = 72$, adjacent is $AC = 54$. So $\tan A = \frac{72}{54} = \frac{4}{3}$, but simplifying $\frac{72}{54} = \frac{4}{3}$, but the options: one of the options is $\frac{72}{54}$ (which is $\frac{4}{3}$), or maybe I made a mistake. Wait, no, wait the options: the fourth option is $\frac{72}{54}$? Wait, the image shows options: first $\frac{54}{72}$, second $\frac{72}{90}$, third $\frac{54}{90}$, fourth $\frac{72}{54}$. Wait, so $\tan A = \frac{72}{54}$, which is one of the options. Wait, but let's check again. $\tan(\theta) = \text{opposite}/\text{adjacent}$. For angle $A$, opposite is $72$, adjacent is $54$. So $\tan A = 72/54 = 4/3$, which is equal to $72/54$. So the correct option is the fourth one? Wait, no, the fourth option is $\frac{72}{54}$? Wait, the options are:

1. $\frac{54}{72}$

1. $\frac{72}{90}$

1. $\frac{54}{90}$

1. $\frac{72}{54}$

So $\tan A = \frac{72}{54}$, so the answer is $\frac{72}{54}$ (simplifies to $\frac{4}{3}$), but among the options, that's the fourth one. Wait, but let's confirm. Yes, because in right triangle, $\tan A = \text{opposite}/\text{adjacent} = 72/54$.

Answer:

$\frac{72}{54}$ (or the option with $\frac{72}{54}$)