To find the difference in length between segments AB and AD in triangles ABC and ADC, we can use the relationships defined in right triangles.
Given:
- Angle \(C\) is a right angle.
- \(AB = AC \cdot \sin(x)\)
- \(AD = AC \cdot \sin(y)\)
- \(AC = 4\) units, \(x = 39^\circ\), and \(y = 58^\circ\).
First, we calculate the lengths of segments \(AB\) and \(AD\).
- Calculate \(AB\):
\[ AB = AC \cdot \sin(x) = 4 \cdot \sin(39^\circ) \]
We can find \(\sin(39^\circ)\):
\[ \sin(39^\circ) \approx 0.6293 \quad (\text{using a calculator}) \]
Now, substituting:
\[ AB \approx 4 \cdot 0.6293 \approx 2.5172 \text{ units} \]
- Calculate \(AD\):
\[ AD = AC \cdot \sin(y) = 4 \cdot \sin(58^\circ) \]
Similarly, find \(\sin(58^\circ)\):
\[ \sin(58^\circ) \approx 0.8480 \quad (\text{using a calculator}) \]
Now, substituting:
\[ AD \approx 4 \cdot 0.8480 \approx 3.3920 \text{ units} \]
- Determine the difference \(AB - AD\):
\[ \text{Difference} = AD - AB = 3.3920 - 2.5172 \approx 0.8748 \text{ units} \]
Now, to find the final answer:
- Round to the nearest hundredth:
The rounded difference is approximately:
\[ 0.87 \text{ units} \]
Based on the provided options, it seems there must be a calculation error as none of the options match. However, if you check to ensure that \(AB < AD\), correct the calculation as needed.
Would you like me to retry the calculations or check other aspects of the problem?