Let us solve the problem step by step. Here's the setup based on the given information:

Problem:

In $\triangle XYZ$:

• $\angle Y = 90^\circ$
• $\angle X = 32^\circ$
• $\angle ZWY = 77^\circ$
• $XW = 5.7$ (segment from $X$ to $W$).

We are tasked with finding the length of $ZY$ to the nearest hundredth.

Step 1: Recognize triangle relationships

From the given diagram, we observe:

1. $\triangle XYZ$ is a right triangle ($\angle Y = 90^\circ$).
2. Using the angles: $\angle Z = 90^\circ - 32^\circ = 58^\circ$ because the sum of angles in any triangle is $180^\circ$.

Step 2: Use trigonometry to find $ZW$

$ZW$ lies along the hypotenuse of right triangle $\triangle XZW$, where:

• $\angle ZWY = 77^\circ$,
• $XW = 5.7$,
• We use the sine rule in $\triangle XZW$.

Using the sine definition: $\sin(77^\circ) = \frac{XW}{ZW}.$ Rearranging for $ZW$: $ZW = \frac{XW}{\sin(77^\circ)} = \frac{5.7}{\sin(77^\circ)}.$ From a calculator: $\sin(77^\circ) \approx 0.97437.$ Thus: $ZW = \frac{5.7}{0.97437} \approx 5.85.$

Step 3: Use Pythagoras or trigonometry to find $ZY$

In $\triangle XYZ$, $ZW$ is the hypotenuse. To find $ZY$, use the sine definition in $\triangle XYZ$: $\sin(32^\circ) = \frac{ZY}{ZW}.$ Rearranging for $ZY$: $ZY = ZW \cdot \sin(32^\circ).$ From a calculator: $\sin(32^\circ) \approx 0.52992.$ Thus: $ZY = 5.85 \cdot 0.52992 \approx 3.10.$

Final Answer:

The length of $ZY$ is approximately 3.10 (to the nearest hundredth).

Would you like further details on any step, or have additional questions?
Here are 5 related questions to deepen understanding:

1. How can you use the cosine function to solve for $ZX$?
2. What is the relationship between $\triangle XZW$ and $\triangle XYZ$?
3. Why is it useful to know all the angles in $\triangle XYZ$?
4. How would the solution change if $XW$ were a different length?
5. Can you solve for $ZX$ using the tangent function?

Tip: Always double-check trigonometric calculations to ensure accuracy, especially with angle measures and rounding!