Question

find the perimeter of the shape. first, label the missing side lengths. 6 cm 3 cm 4 cm 3 cm cm cm

Explanation:

Step1: Find the vertical missing side

The total vertical length on the left is \(3 + 3=6\) cm. The right - hand vertical side should be equal to this. The lower part of the right - hand side is part of a rectangle with height 3 cm, so the missing vertical side (the one in the blue box on the right) is \(6 - 3=3\) cm? Wait, no. Wait, let's look at the horizontal and vertical sides. The top horizontal side is 6 cm, the lower horizontal side of the upper rectangle is 6 cm, and the lower rectangle has a horizontal side of 4 cm. So the horizontal side at the bottom (the blue box at the bottom) is \(6 - 4 = 2\) cm. And the vertical side on the right: the total height of the figure. The left side has a height of 3 cm (upper rectangle) and 3 cm (lower rectangle), so total height is \(3 + 3=6\) cm. The right - hand side: the upper part is part of the upper rectangle (height 3 cm), and the lower part is the height of the lower rectangle (3 cm). Wait, maybe a better way: for the horizontal missing side (bottom blue box): the top horizontal length is 6 cm, the lower rectangle has a horizontal length of 4 cm, so the missing horizontal side is \(6 - 4=2\) cm. For the vertical missing side (right blue box): the left side has a vertical length of \(3 + 3 = 6\) cm, and the lower rectangle has a vertical length of 3 cm, so the missing vertical side is \(6 - 3=3\) cm? Wait, no, let's use the concept of perimeter by translating the sides. If we translate the horizontal and vertical sides, the figure can be thought of as a rectangle with length 6 cm and height \(3 + 3=6\) cm? No, wait, let's list all the sides.

Wait, let's label the sides:

Top: 6 cm

Right - upper: let's see, the upper rectangle has a right - hand side of 3 cm (height of upper rectangle), then the lower part of the right - hand side: the lower rectangle has a height of 3 cm, but the horizontal distance from the lower rectangle to the right - hand side: the top horizontal is 6 cm, the lower rectangle's top horizontal is 4 cm, so the horizontal segment at the bottom (the one we need to find) is \(6 - 4 = 2\) cm. The vertical segment on the right (the one in the blue box) is \(3\) cm? Wait, no, let's do step by step.

First, the horizontal side at the bottom (the blue square at the bottom):

The top horizontal length is 6 cm. The lower rectangle has a horizontal length of 4 cm. So the missing horizontal side (let's call it \(x\)) satisfies \(4 + x=6\), so \(x = 6 - 4=2\) cm.

The vertical side on the right (the blue square on the right):

The left - hand side has a vertical length of \(3+3 = 6\) cm. The lower rectangle has a vertical length of 3 cm. So the missing vertical side (let's call it \(y\)) satisfies \(3 + y=6\), so \(y = 6 - 3=3\) cm.

Now, let's calculate the perimeter.

The sides are:

Top: 6 cm

Right - upper: 3 cm (vertical)

Right - lower: 3 cm (vertical)

Bottom: 6 cm (wait, no, the bottom has two parts: 4 cm and 2 cm, so total bottom length is \(4 + 2=6\) cm)

Left - lower: 3 cm (vertical)

Left - upper: 3 cm (vertical)

Wait, no, let's list all the outer sides:

1. Top: 6 cm

2. Right - upper vertical: 3 cm

3. Right - lower vertical: 3 cm

4. Bottom: \(4 + 2=6\) cm

5. Left - lower vertical: 3 cm

6. Left - upper vertical: 3 cm

7. The horizontal side between the two rectangles: wait, no, when calculating perimeter, we only consider the outer sides. Wait, maybe a better approach: translate the indentation. The horizontal indentation (the 4 cm side) can be translated to the right, and the vertical indentation can be translated up. So the figure is equivalent to a rectangle with length 6 cm and height \(3 + 3=6\) cm? No, wait, the height is \(3+3 = 6\) cm and length 6 cm? Then perimeter would be \(2\times(6 + 6)=24\) cm? Wait, no, let's check with the missing sides.

Wait, let's find the missing sides first:

Horizontal missing side (bottom): \(6 - 4=2\) cm

Vertical missing side (right): \(3\) cm (since the left side has two 3 - cm segments, total 6 cm, and the lower part is 3 cm, so upper part is \(6 - 3 = 3\) cm)

Now, let's list all the outer sides:

- Top: 6 cm

- Right - upper vertical: 3 cm

- Right - lower vertical: 3 cm

- Bottom: \(4 + 2=6\) cm

- Left - lower vertical: 3 cm

- Left - upper vertical: 3 cm

- The horizontal side between the two rectangles: no, that's an inner side, not part of the perimeter. Wait, no, the perimeter is the sum of all outer sides.

Wait, let's count the sides:

Top: 6 cm

Right - upper: 3 cm (vertical)

Right - middle horizontal: 2 cm (wait, no, I think I made a mistake. Let's draw the figure mentally:

Upper rectangle: length 6 cm, height 3 cm.

Lower rectangle: length 4 cm, height 3 cm, attached to the bottom of the upper rectangle but indented by \(6 - 4 = 2\) cm on the right.

So the outer sides:

Top: 6 cm

Right - upper: 3 cm (height of upper rectangle)

Right - vertical: 3 cm (height of lower rectangle)

Bottom: 4 cm (length of lower rectangle) + 2 cm (the indentation on the right)

Left - vertical: 3 cm (height of upper rectangle) + 3 cm (height of lower rectangle)

Left - top horizontal: 6 cm? No, no. Wait, the left side is two vertical segments: 3 cm (upper) and 3 cm (lower), total 6 cm.

The top is 6 cm (horizontal).

The right side: upper vertical segment (3 cm, same as left upper), then a horizontal segment to the left (2 cm, since \(6 - 4=2\)), then a vertical segment down (3 cm, same as left lower), then a horizontal segment to the right (4 cm), then a vertical segment up (3 cm)? No, this is getting confusing. Let's use the method of adding all outer sides:

Sides:

1. Top: 6 cm

2. Right - upper vertical: 3 cm

3. Right - horizontal (the small segment on the right, between the two verticals): 2 cm (since \(6 - 4 = 2\))

4. Right - lower vertical: 3 cm

5. Bottom: 4 cm

6. Left - lower vertical: 3 cm

7. Left - horizontal: 6 cm? No, left - horizontal is the bottom left? Wait, no, the left side is two verticals: 3 cm (upper) and 3 cm (lower). The bottom has a horizontal segment of 4 cm (lower rectangle) and 2 cm (the indentation). The top has 6 cm.

Wait, let's list all the outer edges with their lengths:

- Top: 6 cm (horizontal)

- Right - upper: 3 cm (vertical)

- Right - middle: 2 cm (horizontal, right to left)

- Right - lower: 3 cm (vertical, down)

- Bottom: 4 cm (horizontal, left to right)

- Left - lower: 3 cm (vertical, up)

- Left - middle: 6 cm? No, left - middle is horizontal? Wait, no, the left side is vertical. The top horizontal is 6 cm, the bottom horizontal is \(4 + 2=6\) cm. The left vertical is \(3 + 3=6\) cm. The right vertical is \(3 + 3=6\) cm. Wait, that can't be. Wait, maybe the figure is a rectangle with length 6 cm and height \(3+3 = 6\) cm, but with a notch of a rectangle with length 2 cm and height 3 cm? No, the notch is a rectangle with length \(6 - 4=2\) cm and height 3 cm. Then the perimeter of the original figure (without the notch) would be \(2\times(6 + 6)=24\) cm, and the notch adds two sides of length 3 cm (since we remove a rectangle, we subtract two sides but add two sides of the same length? Wait, no, when you have a notch, the perimeter increases by twice the height of the notch. Wait, the notch has a height of 3 cm, so we add \(2\times3 = 6\) cm? No, let's calculate the perimeter correctly.

Wait, let's find all the side lengths:

- Top: 6 cm

- Right - upper vertical: 3 cm

- Right - horizontal (the small segment on the right, between the two verticals): 2 cm (because \(6 - 4 = 2\))

- Right - lower vertical: 3 cm

- Bottom: 4 cm

- Left - lower vertical: 3 cm

- Left - horizontal (the bottom left horizontal? No, left is vertical. Wait, the left side has two vertical segments: 3 cm (upper) and 3 cm (lower), total 6 cm.

- Top - left horizontal: 6 cm? No, I think I'm overcomplicating. Let's use the missing sides we found: horizontal missing (bottom) is 2 cm, vertical missing (right) is 3 cm.

Now, sum all the sides:

Top: 6

Right - upper vertical: 3

Right - horizontal (the 2 cm? No, wait, the right - hand side: after the upper vertical (3 cm), we have a horizontal segment to the left (2 cm), then a vertical segment down (3 cm), then a horizontal segment to the right (4 cm), then a vertical segment up (3 cm), then a horizontal segment to the left (6 cm)? No, this is wrong.

Wait, let's start over. The figure is composed of two rectangles: upper rectangle (length 6 cm, height 3 cm) and lower rectangle (length 4 cm, height 3 cm), attached such that the lower rectangle is indented by \(6 - 4 = 2\) cm on the right.

To find the perimeter, we can calculate the perimeter of the two rectangles and subtract the overlapping sides (the sides where they are attached, since they are internal and not part of the perimeter).

Perimeter of upper rectangle: \(2\times(6 + 3)=18\) cm

Perimeter of lower rectangle: \(2\times(4 + 3)=14\) cm

Overlapping sides: the side where they are attached is a vertical side of length 3 cm (since the lower rectangle is attached to the upper rectangle, the overlapping side is the top side of the lower rectangle, which is 4 cm? No, no, the overlapping is a vertical side? Wait, no, the upper rectangle has a bottom side of 6 cm, and the lower rectangle has a top side of 4 cm, so the overlapping horizontal side is 4 cm (but it's internal, so we subtract 2 times 4 cm? No, when two rectangles are attached, the perimeter is the sum of the perimeters minus 2 times the length of the overlapping side (because the overlapping side is counted twice in the sum of perimeters, but it's internal, so we subtract 2 times it).

Wait, upper rectangle perimeter: \(2\times(6 + 3)=18\)

Lower rectangle perimeter: \(2\times(4 + 3)=14\)

Overlapping side length: 4 cm (the top side of the lower rectangle, which is attached to the bottom side of the upper rectangle)

So total perimeter: \(18+14 - 2\times4=18 + 14-8 = 24\) cm.

Alternatively, by translating the sides: if we move the 2 cm horizontal segment (the indentation) to the right and the 3 cm vertical segment (the indentation) up, the figure becomes a rectangle with length 6 cm and height \(3 + 3=6\) cm. Then the perimeter is \(2\times(6 + 6)=24\) cm.

Step2: Confirm the missing sides

The horizontal missing side (bottom blue box) is \(6 - 4 = 2\) cm.

The vertical missing side (right blue box) is 3 cm (since the total vertical length on the left is \(3+3 = 6\) cm, and the lower part of the right - hand side is 3 cm, so the upper part is \(6 - 3=3\) cm).

Now, to find the perimeter, we can use the formula for the perimeter of a rectangle (after translating the sides) with length \(l = 6\) cm and width \(w=3 + 3 = 6\) cm. The perimeter \(P=2\times(l + w)=2\times(6 + 6)=24\) cm.

Answer:

The perimeter of the shape is \(\boldsymbol{24}\) cm. The missing horizontal side (bottom) is 2 cm and the missing vertical side (right) is 3 cm.