In `/_\ABC`, right angled at A, `AC = 12cm` and `BC = 15cm` The value of `tanB` is :

Options

A. `(12)/(9)`
B. `(9)/(12)`
C. `(15)/(12)`
D. `(15)/(9)`

Answer

To find the value of \( \tan B \) in triangle \( \triangle ABC \) where \( A \) is the right angle, \( AC = 12 \, \text{cm} \), and \( BC = 15 \, \text{cm} \), we can follow these steps:

### Step 1: Identify the sides of the triangle
In triangle \( ABC \):
- \( AC \) is the side opposite angle \( B \) (perpendicular).
- \( AB \) is the base (adjacent to angle \( B \)).
- \( BC \) is the hypotenuse.

### Step 2: Use the Pythagorean theorem to find \( AB \)
According to the Pythagorean theorem:
\[ BC^2 = AB^2 + AC^2 \]
Substituting the known values:
\[ 15^2 = AB^2 + 12^2 \]
Calculating the squares:
\[ 225 = AB^2 + 144 \]

### Step 3: Solve for \( AB^2 \)
Rearranging the equation gives:
\[ AB^2 = 225 - 144 \]
Calculating the right side:
\[ AB^2 = 81 \]

### Step 4: Find \( AB \)
Taking the square root of both sides:
\[ AB = \sqrt{81} = 9 \, \text{cm} \]

### Step 5: Calculate \( \tan B \)
The formula for \( \tan B \) is:
\[ \tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{AC}{AB} \]
Substituting the values we found:
\[ \tan B = \frac{12}{9} \]
Simplifying the fraction:
\[ \tan B = \frac{4}{3} \]

### Final Answer
Thus, the value of \( \tan B \) is \( \frac{4}{3} \).

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