Question

find the perimeter of each figure. dimensions are in inches. all angles are right angles. (5) figure with dimensions 25, 25, 14, 31 (6) figure with dimensions 25, 12, 17, 31

Explanation:

Step1: Analyze the figure (5)

The figure can be considered as a rectangle with a missing part, but for perimeter, we can use the concept of translating the sides. The horizontal sides: top is 25, bottom is 14 + (25 - 14)? Wait, no, better to use the method of adding all outer sides. Alternatively, notice that the perimeter of such a "notched" rectangle is equal to the perimeter of the larger rectangle that would enclose it. Wait, let's list the sides. The vertical sides: left side? Wait, the figure has height 31, and a notch of 25 in height? Wait, maybe better to calculate each side. Let's see: the top horizontal side is 25, then a vertical side down 25, then a horizontal side right 14, then a vertical side up (31 - 25) = 6, then a horizontal side left (25 - 14) = 11? No, that's complicated. Wait, actually, for a figure with right angles, the perimeter can be calculated by adding all the outer lengths. Alternatively, we can use the formula for the perimeter of a rectangle with a notch: the perimeter is equal to the perimeter of the larger rectangle plus twice the length of the notch's horizontal or vertical sides. Wait, maybe for figure 5: let's consider the outer rectangle. The total height is 31, total width is 25. But there's a notch: the horizontal length of the notch is (25 - 14) = 11, and vertical length is 25? Wait, no, maybe I'm overcomplicating. Let's use the method of translating the sides. For the horizontal sides: the top is 25, the bottom is 14 + (25 - 14) = 25? No, wait, the bottom horizontal side is 14, and then there's a horizontal side above the notch? Wait, maybe the correct way is: the perimeter is calculated by adding all the outer edges. Let's list all the sides:

Top: 25

Right vertical: 31

Bottom: 14 + (25 - 14) = 25? No, bottom is 14, then a horizontal side to the left (25 - 14) = 11? Wait, no, let's draw it mentally: the figure is like a rectangle with length 25 and height 31, but with a square or rectangle cut out from the bottom left? Wait, no, the given dimensions: 25 (top), 25 (left vertical), 14 (bottom horizontal), 31 (right vertical). Wait, maybe the figure is composed of a larger rectangle and a smaller rectangle. Wait, perhaps the perimeter can be found by considering that the horizontal sides sum to 2*(25 + 14)? No, that doesn't make sense. Wait, another approach: for any polygon with right angles, the perimeter is 2*(sum of the horizontal lengths + sum of the vertical lengths). Let's calculate the horizontal lengths: the top horizontal is 25, the bottom horizontal is 14, and then there's a horizontal segment on the left (25 - 14) = 11. Wait, no, horizontal sides: when moving around the figure, the total horizontal distance is 25 (right) + 14 (left) + (25 - 14) (right) + 25 (left)? No, I'm confused. Wait, maybe the correct way is: the perimeter is 2*(25 + 31) + 2*25? No, that would be if there's a notch. Wait, no, let's look at figure 5: the vertical sides: the right side is 31, the left side has a vertical segment of 25, then a vertical segment of (31 - 25) = 6, and then another vertical segment? Wait, maybe I should use the formula for perimeter of a rectangle with a rectangular notch. The perimeter of the original rectangle (without notch) is 2*(length + width). Then, if we cut out a rectangle from the corner, the perimeter increases by twice the length of the notch's horizontal or vertical side. Wait, in figure 5, the notch is a rectangle with width (25 - 14) = 11 and height 25? No, height of the notch is 25, but the total height is 31, so the remaining height is 31 - 25 = 6. Wait, maybe the notch is a rectangle with width 11 (25 - 14) and height 25. Then, the perimeter of the original rectangle (25 x 31) is 2*(25 + 31) = 2*56 = 112. Then, we cut out a rectangle of width 11 and height 25, but when we cut it out, we remove two sides (11 and 25) but add two sides (11 and 25), so the perimeter remains the same? No, that can't be. Wait, no, if we cut out a rectangle from the middle, not the corner, then the perimeter increases. Wait, maybe I'm looking at the figure wrong. Let's try figure 6 first. Figure 6: dimensions 25 (top), 31 (right), 17 (bottom), 12 (left vertical). Wait, similar approach.

Wait, maybe the correct way is: for each figure, the perimeter is calculated by adding all the outer sides. Let's do figure 5:

Sides:

Top: 25

Right: 31

Bottom: 14

Left vertical (from bottom to notch): 31 - 25 = 6

Horizontal (from notch to left): 25 - 14 = 11

Vertical (from notch to top): 25

So adding all these: 25 (top) + 31 (right) + 14 (bottom) + 6 (left vertical) + 11 (horizontal) + 25 (vertical) = 25 + 31 = 56; 56 + 14 = 70; 70 + 6 = 76; 76 + 11 = 87; 87 + 25 = 112. Wait, that's 112? But that's the perimeter of a rectangle with length 25 and width 31: 2*(25 + 31) = 112. So maybe the notch doesn't affect the perimeter? Wait, that's because when you cut out a rectangle from the corner, the perimeter remains the same as the original rectangle. Wait, no, if you cut out a rectangle from the middle, the perimeter increases. But in this case, maybe the notch is at the corner. Wait, maybe figure 5 is a rectangle with length 25 and width 31, with a square of side 14 cut out from the bottom left? No, that doesn't make sense. Wait, maybe the correct perimeter for figure 5 is 2*(25 + 31) = 112. Let's check figure 6: dimensions 25 (top), 31 (right), 17 (bottom), 12 (left vertical). Let's calculate the perimeter. Using the same logic: the total width is 25, total height is 31. The notch has width 12 and height (31 - 25)? No, wait, the bottom horizontal side is 17, top is 25, so the horizontal difference is 25 - 17 = 8, and vertical difference is 31 - 12 = 19? No, better to use the rectangle perimeter. Wait, 2*(25 + 31) = 112? No, that can't be. Wait, maybe I made a mistake. Let's try again.

Wait, the problem says "All angles are right angles". So for a figure with right angles, the perimeter can be calculated by adding the sum of all horizontal sides and twice the sum of all vertical sides, or vice versa. Wait, horizontal sides: for figure 5, the top horizontal is 25, the bottom horizontal is 14, and then there's a horizontal side above the notch: (25 - 14) = 11. So total horizontal length: 25 + 14 + 11 = 50? No, that's not right. Wait, no, in a polygon with right angles, the sum of the horizontal sides (left and right) is equal, and the sum of the vertical sides (top and bottom) is equal. Wait, actually, for any closed polygon with right angles, the perimeter is 2*(sum of the horizontal lengths + sum of the vertical lengths). Wait, let's take figure 5:

Horizontal sides: top (25), bottom (14), and the horizontal side of the notch (25 - 14) = 11. So total horizontal length: 25 + 14 + 11 = 50? No, that's the sum of all horizontal sides, but since it's a closed polygon, the sum of the horizontal sides (leftward and rightward) should be equal. Wait, maybe I'm overcomplicating. Let's use the formula for the perimeter of a rectangle with a rectangular cutout. If you have a rectangle of length L and width W, and you cut out a rectangle of length l and width w from the corner, the perimeter remains 2*(L + W). But if you cut out a rectangle from the middle, the perimeter becomes 2*(L + W) + 2*l or 2*w, depending on the cut.

Wait, looking at figure 5: the dimensions are 25 (top), 25 (left vertical), 14 (bottom horizontal), 31 (right vertical). So the total height is 31, total width is 25. The notch is a rectangle with width (25 - 14) = 11 and height 25. Wait, but the height of the notch is 25, and the total height is 31, so the remaining height is 31 - 25 = 6. So when we cut out the notch, we remove a horizontal side of 11 and a vertical side of 25, but we add a horizontal side of 11 and a vertical side of 25, so the perimeter remains the same as the original rectangle. Wait, original rectangle perimeter: 2*(25 + 31) = 2*56 = 112. So figure 5 perimeter is 112.

Now figure 6: dimensions 25 (top), 31 (right), 17 (bottom), 12 (left vertical). Total width is 25, total height is 31. The notch has width (25 - 17) = 8 and height (31 - 12) = 19? No, wait, the bottom horizontal is 17, left vertical is 12. So the notch is a rectangle with width 12 and height (31 - 25) = 6? No, maybe the same logic: the perimeter is 2*(25 + 31) = 112? Wait, no, that can't be. Wait, maybe I'm misinterpreting the figures. Let's look at the dimensions again. For figure 5: top 25, left vertical 25, bottom 14, right vertical 31. So the horizontal sides: top 25, bottom 14, and the horizontal side between the two vertical sides (25 - 14) = 11. Vertical sides: right 31, left 25 + (31 - 25) = 31. So total perimeter: 25 (top) + 31 (right) + 14 (bottom) + 31 (left) + 11 (middle horizontal) + 25 (middle vertical)? No, that's not right. Wait, no, in a right-angled polygon, the sum of the horizontal sides (left and right) is equal, and the sum of the vertical sides (top and bottom) is equal. Wait, horizontal sides: when moving around the figure, the total horizontal distance to the right is equal to the total horizontal distance to the left. Similarly for vertical.

For figure 5:

Right horizontal movement: 25 (top)

Left horizontal movement: 14 (bottom) + (25 - 14) = 25 (so total left horizontal is 25, same as right)

Vertical movement up: 31 (right)

Vertical movement down: 25 (left) + (31 - 25) = 31 (so total down vertical is 31, same as up)

So perimeter is 2*(25 + 31) = 112.

For figure 6:

Top horizontal: 25

Bottom horizontal: 17 + (25 - 17) = 25 (so total horizontal right and left are 25)

Right vertical: 31

Left vertical: 12 + (31 - 12) = 31 (so total vertical up and down are 31)

Thus perimeter is 2*(25 + 31) = 112? Wait, that can't be, both figures have the same perimeter? That seems odd. Wait, maybe the dimensions are different. Wait, figure 5: left vertical is 25, right vertical is 31, so the difference is 6. Figure 6: left vertical is 12, right vertical is 31, difference is 19. Wait, maybe my approach is wrong. Let's calculate figure 6 manually.

Figure 6:

Top: 25

Right: 31

Bottom: 17

Left vertical (from bottom to notch): 12

Horizontal (from notch to left): 25 - 17 = 8

Vertical (from notch to top): 31 - 12 = 19

So adding all sides: 25 (top) + 31 (right) + 17 (bottom) + 12 (left vertical) + 8 (horizontal) + 19 (vertical) = 25 + 31 = 56; 56 + 17 = 73; 73 + 12 = 85; 85 + 8 = 93; 93 + 19 = 112. Oh! So both figures have perimeter 112. That's because the sum of the horizontal sides is 25 + 17 + 8 = 50, and sum of vertical sides is 31 + 12 + 19 = 62? No, wait, perimeter is sum of all sides: 25 + 31 + 17 + 12 + 8 + 19 = 112. Yes, same as figure 5. So the perimeter for both figures is 112 inches.

Step2: Verify the calculation

For figure 5:

Sides: 25 (top), 31 (right), 14 (bottom), 6 (left vertical, 31 - 25), 11 (horizontal, 25 - 14), 25 (vertical). Sum: 25 + 31 + 14 + 6 + 11 + 25 = 112.

For figure 6:

Sides: 25 (top), 31 (right), 17 (bottom), 12 (left vertical), 8 (horizontal, 25 - 17), 19 (vertical, 31 - 12). Sum: 25 + 31 + 17 + 12 + 8 + 19 = 112.

So both figures have a perimeter of 112 inches.

Answer:

For figure 5: $\boxed{112}$ inches
For figure 6: $\boxed{112}$ inches