To solve the problem shown in the image:

Part (a): Find the lengths of the parallel sides of the trapezium

The trapezium has two parallel sides:

1. The shorter parallel side is given as 24 cm.
2. The longer parallel side is the bottom side, which measures 70 cm.

Answer for (a):
The lengths of the parallel sides are:

• Shorter parallel side: 24 cm
• Longer parallel side: 70 cm

Part (b): Find the perpendicular height of the trapezium

We can use the area of a triangle formula or basic trigonometry to find the perpendicular height.

Step 1: Height of the Left Triangle

The left side of the trapezium forms a triangle with:

• Hypotenuse = 26 cm (left slanting side)
• Base = 15 cm (part of the bottom parallel side)

Using Pythagoras' theorem: $\text{Height}^2 = \text{Hypotenuse}^2 - \text{Base}^2$ $\text{Height}^2 = 26^2 - 15^2 = 676 - 225 = 451$ $\text{Height} = \sqrt{451} \approx 21.24 \, \text{cm}$

So, the height of the left triangle is approximately 21.24 cm.

Step 2: Verification with Other Triangle (Optional)

The height of the trapezium should be consistent with the second triangle, verifying the perpendicular drop is the same across the trapezium.

Final Answer:

• Lengths of parallel sides: 24 cm and 70 cm
• Perpendicular height: Approximately 21.24 cm.

If you have more questions or need further clarification, let me know!

Expanding the Problem:

1. How can you use the area formula of a trapezium to double-check the height?
2. What other methods exist to calculate height if angles are given?
3. How does the perpendicular height affect the area of a trapezium?
4. Can the height differ depending on which triangle we use to calculate it?
5. What role does Pythagoras’ theorem play in solving trapezium problems?

Tip: Always verify the height calculation with both triangles in a trapezium for accuracy.