From the provided image, the problem involves a trapezium, and the goal is to calculate the length of $x$ in centimeters, given certain dimensions.

Step-by-Step Solution:

1. Trapezium Geometry Analysis:

   • A trapezium is a quadrilateral with one pair of parallel sides.
   • In this problem, a triangle is part of the trapezium, and we are solving for $x$ using given dimensions.

2. Key Measurements:

   • The height (perpendicular distance): 17 cm.
   • The base: $5 \, \text{cm} + x$.
   • The given shorter height from the triangle is $3 \, \text{cm}$.

3. Apply Pythagoras' Theorem:

   • For the triangle on the left, the right-angled triangle dimensions are:
     • Height: $17 - 3 = 14 \, \text{cm}$.
     • Base: $x$.

   Using the Pythagorean theorem: $x^2 + 14^2 = 17^2$

4. Solve for $x$: $x^2 = 17^2 - 14^2$ $x^2 = 289 - 196$ $x^2 = 93$ $x = \sqrt{93}$ $x \approx 9.6 \, \text{cm (to 1 decimal place)}.$

Final Answer:

The length of $x$ is approximately 9.6 cm.

Would you like a deeper explanation of any step or additional help? Here are 5 follow-up questions you might explore:

1. What other methods could be used to solve for $x$ in a right-angled triangle?
2. How can the Pythagorean theorem be derived geometrically?
3. What is the difference between a trapezium and other quadrilaterals?
4. How do we ensure precision when calculating square roots?
5. Can you calculate the area of the trapezium using the dimensions provided?

Tip: Always double-check your substitution in formulas to avoid small errors in multi-step problems!