In `triangleXYZ` measure of angle Y is `90^@`. If sinX = 4/5, and XY = 6cm, then what is the length (in cm) of side YZ?

Options

A. 10
B. 5
C. 8
D. 4

Answer

To solve the problem step by step, we will use the properties of right triangles and the given information.

### Step 1: Understand the triangle and given information
We have a right triangle XYZ where angle Y is 90 degrees. We know:
- \( \sin X = \frac{4}{5} \)
- \( XY = 6 \, \text{cm} \)

### Step 2: Relate sine to the sides of the triangle
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse. Therefore, we can write:
\[ \sin X = \frac{YZ}{ZX} \]
Given that \( \sin X = \frac{4}{5} \), we can express this as:
\[ \frac{YZ}{ZX} = \frac{4}{5} \]
Let \( YZ = 4k \) and \( ZX = 5k \) for some value \( k \).

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ ZX^2 = XY^2 + YZ^2 \]
Substituting the values we have:
\[ (5k)^2 = (6)^2 + (4k)^2 \]
This simplifies to:
\[ 25k^2 = 36 + 16k^2 \]

### Step 4: Solve for \( k \)
Rearranging the equation gives:
\[ 25k^2 - 16k^2 = 36 \]
\[ 9k^2 = 36 \]
Dividing both sides by 9:
\[ k^2 = 4 \]
Taking the square root:
\[ k = 2 \]

### Step 5: Find the length of side \( YZ \)
Now that we have \( k \), we can find \( YZ \):
\[ YZ = 4k = 4 \times 2 = 8 \, \text{cm} \]

### Final Answer
The length of side \( YZ \) is \( 8 \, \text{cm} \).

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