In `DeltaXYZ` measure of angle Y is `90^(@)` .If `cosecX=(17)/(15),andXY=4cm` , what is the length (n cm .) of side YZ ?

Options

A. 7.5
B. 8.5
C. 5
D. 6

Answer

To solve the problem step-by-step, we will use the information provided about triangle XYZ, where angle Y is 90 degrees, and we know the cosecant of angle X and the length of side XY.

### Step 1: Understand the triangle and given values
In triangle XYZ:
- Angle Y = 90°
- Cosec X = 17/15
- Length of side XY = 4 cm

### Step 2: Relate cosecant to the sides of the triangle
Cosecant is defined as the ratio of the hypotenuse to the opposite side. Therefore, we can write:
\[ \text{cosec } X = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{YZ}{XY} \]
From the given information:
\[ \frac{YZ}{XY} = \frac{17}{15} \]

### Step 3: Substitute the known value of XY
We know that XY = 4 cm. Substituting this value into the equation gives:
\[ \frac{YZ}{4} = \frac{17}{15} \]

### Step 4: Solve for YZ
To find YZ, we can cross-multiply:
\[ YZ \cdot 15 = 17 \cdot 4 \]
Calculating the right side:
\[ YZ \cdot 15 = 68 \]
Now, divide both sides by 15 to isolate YZ:
\[ YZ = \frac{68}{15} \]

### Step 5: Calculate the value of YZ
Now we can calculate the value:
\[ YZ = \frac{68}{15} \approx 4.53 \text{ cm} \]

### Step 6: Finalize the answer
Thus, the length of side YZ is approximately 4.53 cm.

### Summary of the solution:
- Cosec X relates the sides of the triangle.
- Substituting the known value of XY allows us to find YZ.
- The final calculated length of YZ is approximately 4.53 cm.