`triangleXYZ` is right angled at y and `angleZ=30^(@)` what is the length of yz if zx=9 cm

Options

A. `6sqrt(3)`
B. `9sqrt(3)/(2)`
C. `3sqrt(3)`
D. 18

Answer

To solve the problem, we will use the properties of a right-angled triangle. Given that triangle XYZ is right-angled at Y and angle Z is 30 degrees, we need to find the length of side YZ when the hypotenuse ZX is 9 cm.

### Step-by-Step Solution:

1. **Identify the triangle and given values**:
We have triangle XYZ where:
- Angle Y is 90 degrees (right angle)
- Angle Z is 30 degrees
- Hypotenuse ZX = 9 cm

2. **Determine the angles**:
Since the sum of angles in a triangle is 180 degrees:
- Angle X = 180° - Angle Y - Angle Z
- Angle X = 180° - 90° - 30° = 60°

3. **Use trigonometric ratios**:
In a right-angled triangle, we can use the sine and cosine functions to find the lengths of the sides. We know:
- The sine of an angle is defined as the opposite side over the hypotenuse.

For angle Z (30 degrees):
\[ \sin(Z) = \frac{YZ}{ZX} \]

Substituting the known values:
\[ \sin(30°) = \frac{YZ}{9} \]

4. **Calculate sin(30°)**:
We know that:
\[ \sin(30°) = \frac{1}{2} \]

5. **Set up the equation**:
Now we can substitute this value into our equation:
\[ \frac{1}{2} = \frac{YZ}{9} \]

6. **Solve for YZ**:
To find YZ, we can multiply both sides by 9:
\[ YZ = 9 \times \frac{1}{2} = \frac{9}{2} = 4.5 \text{ cm} \]

### Final Answer:
The length of YZ is 4.5 cm.