To find the measure of segment \( YX \) given that \( \sin f^\circ = \frac{8}{9} \) and the measure of segment \( YW = 24 \) units, we can use the definition of the sine function in a right triangle.
In triangle \( XYW \), where angle \( W \) is the right angle, we have:
- \( YW \) is the side opposite angle \( X \),
- \( YX \) is the hypotenuse,
- and \( XY \) is the side adjacent to angle \( X \).
Given:
- \( YW = 24 \) units,
- \( \sin f^\circ = \frac{8}{9} \).
From the definition of sine: \[ \sin f^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{YW}{YX} \] Substituting the known values: \[ \frac{8}{9} = \frac{24}{YX} \]
To solve for \( YX \), we can cross-multiply: \[ 8 \cdot YX = 9 \cdot 24 \] \[ 8 \cdot YX = 216 \] Now, divide both sides by 8: \[ YX = \frac{216}{8} = 27 \]
Thus, the measure of segment \( YX \) is 27 units.
The answer is: 27 units
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