In `DeltaXYZ` measure of angle Y is `90^(@)`.Ifcot X `=(5)/(12)andXY=2.5` cm , then what is the length (in cm .) of side XZ?

Options

A. 6
B. 4
C. 5.6
D. 6.5

Answer

To find the length of side XZ in triangle XYZ where angle Y is 90 degrees, cotangent of angle X is given as \( \frac{5}{12} \), and XY is 2.5 cm, we can follow these steps:

### Step 1: Understand the relationship of cotangent
The cotangent of an angle in a right triangle is defined as the ratio of the adjacent side to the opposite side. Here, for angle X:
\[ \cot X = \frac{\text{adjacent}}{\text{opposite}} = \frac{XY}{YZ} \]
Given that \( \cot X = \frac{5}{12} \), we can assign:
- Adjacent side (XY) = 5 units
- Opposite side (YZ) = 12 units

### Step 2: Set up the ratio based on the given length of XY
Since XY is given as 2.5 cm, we can find the actual lengths of YZ and XZ using the ratio of 5:12.

Let’s denote the unit length as \( k \):
- \( XY = 5k = 2.5 \)
- Therefore, \( k = \frac{2.5}{5} = 0.5 \)

### Step 3: Calculate the length of YZ
Using \( k \):
\[ YZ = 12k = 12 \times 0.5 = 6 \text{ cm} \]

### Step 4: Use the Pythagorean theorem to find XZ
In a right triangle, we can use the Pythagorean theorem:
\[ XZ^2 = XY^2 + YZ^2 \]
Substituting the known lengths:
\[ XZ^2 = (2.5)^2 + (6)^2 \]
Calculating the squares:
\[ XZ^2 = 6.25 + 36 = 42.25 \]
Taking the square root:
\[ XZ = \sqrt{42.25} = 6.5 \text{ cm} \]

### Final Answer
The length of side XZ is \( 6.5 \) cm.

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