To solve for $x$, we can use trigonometric ratios since the triangle involves a right angle ($\angle W$) and the given angle of $52^\circ$. The sides provided and the one to find are associated with the given angle $52^\circ$.

Step 1: Identify sides relative to $\angle T = 52^\circ$:

1. $x$ is the hypotenuse of the triangle.
2. $96$ is the adjacent side to the angle.

Step 2: Select the appropriate trigonometric ratio.

Using the cosine function (since cosine involves adjacent and hypotenuse): $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Here: $\cos(52^\circ) = \frac{96}{x}$

Step 3: Solve for $x$.

Rearranging the equation to solve for $x$: $x = \frac{96}{\cos(52^\circ)}$

Step 4: Calculate $x$.

Using a calculator: $\cos(52^\circ) \approx 0.6157$

So: $x = \frac{96}{0.6157} \approx 155.9$

Final Answer:

$x \approx 155.9$

Would you like further clarification or additional details?

Related Questions:

1. What are the sine and tangent values of $52^\circ$ for this triangle?
2. How would the problem change if the angle $\angle T$ was different, say $45^\circ$?
3. How do you verify your result if $x$ is given, and you need to find another side?
4. Can you solve for $x$ using the Pythagorean Theorem in this problem?
5. What is the relationship between angles and their corresponding side lengths in trigonometric problems?

Tip:

Always double-check your angle mode (degrees or radians) on your calculator when working with trigonometric functions!