In `DeltaXYZ` measure of angle Y is `90^(@)` .If cosec `X=(13)/(12),andXY=1`cm ,then what is the length (in cm.)of side YZ?

Options

A. 2.6
B. 1.5
C. 2.4
D. 2

Answer

To solve the problem step by step, we will follow the given information and apply trigonometric principles.

### Step 1: Understand the triangle and given values
We have a right triangle \( \Delta XYZ \) with \( \angle Y = 90^\circ \). We are given:
- \( \csc X = \frac{13}{12} \)
- \( XY = 1 \, \text{cm} \)

### Step 2: Relate cosecant to the sides of the triangle
The cosecant of an angle in a right triangle is defined as:
\[ \csc X = \frac{\text{Hypotenuse}}{\text{Opposite side to angle X}} \]
In this case:
- Hypotenuse \( YZ = 13k \)
- Opposite side to angle \( X \) (which is \( YZ \)) = \( 12k \)

From the cosecant value, we can deduce:
\[ \csc X = \frac{YZ}{XY} = \frac{13}{12} \]

### Step 3: Determine the lengths of the sides
Since \( XY = 1 \, \text{cm} \), we can set up the ratio:
\[ \frac{YZ}{XY} = \frac{13}{12} \]
Let \( k \) be a common factor. Then:
- \( YZ = 12k \)
- \( XY = 1 \, \text{cm} \)

### Step 4: Use the Pythagorean theorem
According to the Pythagorean theorem:
\[ YZ^2 = XY^2 + XZ^2 \]
Substituting the known values:
\[ (13k)^2 = (1)^2 + (12k)^2 \]
This simplifies to:
\[ 169k^2 = 1 + 144k^2 \]
Rearranging gives:
\[ 169k^2 - 144k^2 = 1 \]
\[ 25k^2 = 1 \]
\[ k^2 = \frac{1}{25} \]
\[ k = \frac{1}{5} \]

### Step 5: Calculate the length of side YZ
Now substituting back to find \( YZ \):
\[ YZ = 12k = 12 \times \frac{1}{5} = \frac{12}{5} \, \text{cm} \]

### Step 6: Final answer
Thus, the length of side \( YZ \) is:
\[ \frac{12}{5} \, \text{cm} = 2.4 \, \text{cm} \]