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The given figure is a trapezium, where we are asked to find the value of \( x \) and the perimeter.

### Step 1: Understand the given dimensions
- One of the parallel sides of the trapezium is \( 7 \, \text{m} \).
- The non-parallel sides are \( 4 \, \text{m} \) each.
- The other parallel side is \( x \, \text{m} \).

### Step 2: Analyze the figure
The two non-parallel sides of length \( 4 \, \text{m} \) form a right triangle with part of the longer parallel side (base of \( 7 \, \text{m} \)).

Using symmetry, the shorter base of the trapezium is evenly divided into two parts, so the length of each part will be \( \frac{7 - x}{2} \, \text{m} \).

### Step 3: Use the Pythagorean Theorem
Since a right triangle is formed by:
- One leg = \( \frac{7 - x}{2} \)
- Another leg = \( 4 \, \text{m} \)
- Hypotenuse = \( 4 \, \text{m} \)

Using the Pythagorean theorem:

\[
\left( \frac{7 - x}{2} \right)^2 + 4^2 = 4^2
\]

Solve this equation for \( x \).

\[
\left( \frac{7 - x}{2} \right)^2 + 16 = 16
\]

Simplifying:

\[
\left( \frac{7 - x}{2} \right)^2 = 0
\]

This gives:

\[
\frac{7 - x}{2} = 0
\]

So:

\[
7 - x = 0
\]

\[
x = 7
\]

### Step 4: Calculate the perimeter
The perimeter \( P \) of the trapezium is the sum of all its sides:

\[
P = 7 + x + 4 + 4 = 7 + 7 + 4 + 4 = 22 \, \text{m}
\]

### Final Answer:
- \( x = 7 \, \text{m} \)
- Perimeter \( P = 22 \, \text{m} \)

Would you like more details or clarifications?

Here are 5 related questions to explore:
1. What is the formula for the area of a trapezium?
2. How does symmetry help in solving trapezium problems?
3. Can the Pythagorean theorem be used for any trapezium?
4. What is the difference between a trapezium and a parallelogram?
5. How would the problem change if the sides were not equal?

**Tip**: Always look for symmetry in geometric shapes to simplify complex problems.

Find the value of x and the perimeter of the trapezium, given the dimensions 7m, 4m, and x meters.

Learn how to find the unknown side x and the perimeter of a trapezium with given sides 7m, 4m, and x. This geometry problem uses the Pythagorean theorem and symmetry to solve for x and calculate the perimeter. Suitable for students in Grades 8-10.

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