Area and Perimeter Worksheet 4 - Practice calculating perimeters and solving for unknown lengths in geometric figures.

Let's solve each problem step by step from the worksheet.

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(1) Find $ b - a $?

We are given a composite shape made of rectangles, with some side lengths labeled:

- The total horizontal length at the bottom is 11 cm.
- The top horizontal segment is split into two parts: 7 cm and a, and there’s another segment b on the left side.
- The vertical sides are:
- Left side: 6 cm
- Right side: 8 cm

Let’s analyze the figure carefully.

This shape appears to be a rectangle with a smaller rectangle removed or extended on top. But since it's a closed shape, we can use the fact that opposite sides are equal in a rectangle.

#### Step-by-step analysis:

The total height on the right side is 8 cm, and on the left side it's 6 cm, so the difference must come from the "step" in the middle.

Looking at the top edge:
- Total top length = 7 cm + a
- But the bottom length is 11 cm → So, the horizontal projection of the left side (with length b) must make up the rest.

Wait — let's think differently.

Actually, look at the horizontal segments:

- Bottom: 11 cm
- Top: 7 cm + a

But for the shape to close properly, the total horizontal length must match.

However, the left side has a vertical segment of 6 cm, and the right side has 8 cm.

So, the difference in vertical height is due to the "step" in the shape.

Now consider the horizontal projections:

- The top part is 7 cm long and is aligned with the upper portion.
- The bottom part is 11 cm long.
- The vertical drop on the left side is 6 cm, and on the right side is 8 cm.

But the key is that the total horizontal width must be consistent.

Let’s reconstruct:

From the diagram:
- The base is 11 cm.
- The top is split: one part is 7 cm, and the other is 'a'.
- On the left, there is a vertical segment of 6 cm, then a horizontal segment of 'b', then upward to meet the top.

But actually, the left side has:
- A vertical segment of 6 cm (bottom),
- Then a horizontal segment of b going right,
- Then a vertical segment of (8 - 6) = 2 cm upward,
- Then a horizontal segment of 7 cm to the right,
- Then down 8 cm to complete the shape.

Wait — better approach:

Let’s look at horizontal alignment.

The total width at the bottom is 11 cm.

On the top, we have:
- A segment of 7 cm on the right,
- And a segment of 'a' on the left (next to 'b').

But also, the left side has a vertical segment of 6 cm, and the right side has 8 cm.

So the horizontal extension on the left must account for the difference in height.

But here's the trick:

Look at the horizontal lines:

- The bottom is 11 cm.
- The top is divided into two parts: 7 cm and 'a'.

But the vertical line between them is not straight — it steps up.

Let’s suppose the shape is like this:

```
_________
| |
| | 8 cm
|______ |
| | 6 cm
| |
|_______|
```

Wait — better:

It's an L-shaped figure? No — it looks more like a rectangular shape with a rectangular notch on the top-left?

No — actually, it seems like a rectangle with a smaller rectangle added on top.

Wait — looking again:

- The right side is 8 cm tall.
- The left side is 6 cm tall.
- So, the top must go up 2 cm higher on the right than on the left.

So the top edge starts at 6 cm height and goes up to 8 cm at the right.

But the top horizontal segment is only 7 cm long, and there's a segment 'a' and 'b' on the left.

Let’s try to use horizontal balance.

Total bottom width = 11 cm.

The top has:
- 7 cm on the right,
- And 'a' on the left (which is connected to 'b').

But the horizontal segment 'b' is likely the overhang on the left.

Wait — perhaps the figure is like this:

Imagine a large rectangle of width 11 cm and height 8 cm.

Then, on the left side, a smaller rectangle of height 6 cm is cut out or attached?

But the left side shows 6 cm, and the right side shows 8 cm.

So the shape is taller on the right.

Thus, the top must be stepped: from the left side (6 cm), it goes up to 8 cm on the right.

So the top edge is broken into two parts:
- One part of 7 cm at the right end (at height 8 cm),
- And the rest is at height 6 cm.

But wait — the top edge is drawn as:
- A horizontal segment of 7 cm,
- Then a vertical segment down?
No — the diagram shows:

From the top-left, it goes:
- Rightward for 'b',
- Then up for 'a',
- Then right for 7 cm,
- Then down 8 cm,
- Then left for 11 cm,
- Then up 6 cm.

Wait — no.

Let me interpret the diagram clearly:

The figure is a polygon with the following sides:

1. Start at bottom-left corner.
2. Go right 11 cm → bottom.
3. Go up 8 cm → right side.
4. Go left 7 cm → top-right.
5. Go down 'a' → vertical segment.
6. Go left 'b' → horizontal segment.
7. Go up 6 cm → back to start.

Wait — but the left side is labeled 6 cm.

So the path should be:

- Start at bottom-left.
- Up 6 cm → left side.
- Then right 'b' → horizontal.
- Then up 'a' → vertical.
- Then right 7 cm → top.
- Then down 8 cm → right side.
- Then left 11 cm → bottom.

Wait — but the bottom is 11 cm, so if you go from right to left 11 cm, that matches.

But now, check horizontal consistency.

From the bottom-left, you go up 6 cm, then right 'b', then up 'a', then right 7 cm, then down 8 cm, then left 11 cm.

So the total horizontal movement:

- Right: 'b' + 7 cm
- Left: 11 cm

For the shape to close, the net horizontal displacement must be zero.

But we’re going right (b + 7), then left 11 cm.

So:
$$
b + 7 = 11 \Rightarrow b = 4
$$

Similarly, vertically:

- Up: 6 cm + a
- Down: 8 cm

So:
$$
6 + a = 8 \Rightarrow a = 2
$$

Now, we are asked: $ b - a = ? $

$$
b - a = 4 - 2 = 2
$$

✔ Answer: (b) 2

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(2) Perimeter of the triangle

Triangle with sides:
- 3 cm, 3 cm, and 4 cm

This is an isosceles triangle.

Perimeter = sum of all sides:
$$
3 + 3 + 4 = 10 \text{ cm}
$$

✔ Answer: (a) 10

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(3) Perimeter of the triangle

Triangle with sides:
- 3 cm, 4 cm, 5 cm

This is a right triangle (since $ 3^2 + 4^2 = 9 + 16 = 25 = 5^2 $)

Perimeter = $ 3 + 4 + 5 = 12 $ cm

✔ Answer: (b) 12

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✔ Final Answers:

1. $ b - a = 2 $ → (b) 2
2. Perimeter = 10 units → (a) 10
3. Perimeter = 12 units → (b) 12

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