In this triangle problem, we are tasked with finding the length of $CE = x$ using the provided right triangle $\triangle CDE$.

We are given:

1. $\angle CDE = 90^\circ$ (a right angle),
2. $\angle C = 17^\circ$,
3. $DE = 1.7$ (adjacent side to angle $C$).

We can use trigonometric functions since it is a right triangle. Specifically, for the given angle $\angle C = 17^\circ$, we use the cosine function:

$\cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$

Step 1: Plugging in known values

• $\theta = 17^\circ$,
• Adjacent side = $DE = 1.7$,
• Hypotenuse = $CE = x$.

So:

$\cos(17^\circ) = \frac{1.7}{x}$

Step 2: Solve for $x$

Rearranging for $x$:

$x = \frac{1.7}{\cos(17^\circ)}$

Step 3: Calculate the value

We calculate $\cos(17^\circ)$ using a calculator:

$\cos(17^\circ) \approx 0.9563$

Substitute this into the equation:

$x = \frac{1.7}{0.9563} \approx 1.78$

Final Answer:

The length of $CE = x$ is approximately 1.78 units.

Do you need a detailed explanation or further verification?

Here are 5 related questions for deeper understanding:

1. How do we use the sine function for similar triangles?
2. What is the tangent function, and when is it used in right triangles?
3. If the angle changes, how would that affect the hypotenuse?
4. Can you explain why trigonometric ratios depend on angles?
5. How do we find missing angles using trigonometric ratios?

Tip: Always use trigonometric functions (sine, cosine, or tangent) based on which sides you are given (opposite, adjacent, or hypotenuse).