`triangleXYZ `is right angled at Y. If tan X = 24/7, then what is the value of cot Z?

Options

A. `25//7`
B. `24//25`
C. `7//24`
D. `24//7`

Answer

To solve the problem, we need to find the value of cot Z in triangle XYZ, which is right-angled at Y. Given that tan X = 24/7, we can follow these steps:

### Step 1: Understand the triangle
In triangle XYZ, angle Y is the right angle. Therefore, angles X and Z are complementary, meaning that X + Z = 90 degrees.

### Step 2: Use the definition of tangent
We know that:
\[ \tan X = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\text{Perpendicular}}{\text{Base}} \]
Given that \(\tan X = \frac{24}{7}\), we can assign:
- Opposite side (to angle X) = 24λ (where λ is a scaling factor)
- Adjacent side (to angle X) = 7λ

### Step 3: Find the sides of the triangle
Since angle Y is 90 degrees, we can find the opposite and adjacent sides for angle Z:
- Opposite side (to angle Z) = 7λ (which is the adjacent side for angle X)
- Adjacent side (to angle Z) = 24λ (which is the opposite side for angle X)

### Step 4: Calculate \(\tan Z\)
Using the definition of tangent for angle Z:
\[ \tan Z = \frac{\text{Opposite to Z}}{\text{Adjacent to Z}} = \frac{7λ}{24λ} \]
The λ cancels out:
\[ \tan Z = \frac{7}{24} \]

### Step 5: Find \(\cot Z\)
The cotangent of an angle is the reciprocal of the tangent:
\[ \cot Z = \frac{1}{\tan Z} = \frac{1}{\frac{7}{24}} = \frac{24}{7} \]

### Conclusion
Thus, the value of \(\cot Z\) is \(\frac{24}{7}\).