To solve for the angle $x$, we can use the cosine rule for a right triangle:

$\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}}$

From the diagram:

• The adjacent side (base) is $25.7 \, \text{mm}$.
• The opposite side is $11.8 \, \text{mm}$.

Step 1: Calculate the hypotenuse

Using the Pythagorean theorem: $c = \sqrt{a^2 + b^2}$ where $a = 11.8 \, \text{mm}$ and $b = 25.7 \, \text{mm}$.

$c = \sqrt{(11.8)^2 + (25.7)^2}$

Calculating: $c = \sqrt{139.24 + 660.49} = \sqrt{799.73} \approx 28.3 \, \text{mm (to 1 d.p.)}$

Step 2: Find $\cos(x)$

$\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{25.7}{28.3}$

Calculating: $\cos(x) \approx 0.908$

Step 3: Solve for $x$ using the inverse cosine function

$x = \cos^{-1}(0.908)$

Using a calculator: $x \approx 25.2^\circ$

Final Answer:

The size of angle $x$ is 25.2° (to 1 decimal place).

Do you need further clarification on any step? Let me know!

Related Questions:

1. How would you calculate the sine and tangent of $x$?
2. What is the area of this triangle?
3. Can you solve for the opposite angle of $x$?
4. How does the Pythagorean theorem work for this triangle?
5. How would the solution change if one side length were different?

Tip: Always double-check which trigonometric function (sine, cosine, or tangent) applies to the sides of the triangle you are working with.