Question

f. find the perimeter of the designs below.

1.
an l-shaped polygon with a vertical left side of length \\(70\text{ cm}\\), a horizontal bottom side of length \\(100\text{ cm}\\), a vertical right side of length \\(35\text{ cm}\\), and an inner horizontal step of length \\(40\text{ cm}\\).

2.
an l-shaped polygon with a vertical right side of length \\(115\text{ cm}\\), a horizontal bottom side of length \\(75\text{ cm}\\), a vertical inner step of length \\(45\text{ cm}\\), and a horizontal top side of length \\(36\text{ cm}\\).

3.
an l-shaped polygon with a horizontal top side of length \\(125\text{ cm}\\), a vertical left side of length \\(64\text{ cm}\\), a horizontal bottom-left side of length \\(75\text{ cm}\\), and a vertical inner step of length \\(32\text{ cm}\\).

4.
an l-shaped polygon with a horizontal top side of length \\(275\text{ cm}\\), a vertical left side of length \\(36\text{ cm}\\), a vertical inner step of length \\(36\text{ cm}\\), and a horizontal bottom-right side of length \\(190\text{ cm}\\).

Explanation:

🆕 New Concept Discovered: Perimeter of L-shaped Polygons
Finding the total boundary length of composite shapes

To find the perimeter of an L-shaped polygon, we need to sum the lengths of all its outer sides. If some side lengths are missing, we can find them by comparing parallel horizontal or vertical segments.

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Step 1: Solve Problem 1

Let's identify all 6 sides of the first shape:

• Left vertical side: \( 70\text{ cm} \)
• Bottom horizontal side: \( 100\text{ cm} \)
• Right vertical side: \( 35\text{ cm} \)
• Inner horizontal step: \( 40\text{ cm} \)
• Top horizontal side: We can find this by looking at the horizontal segments. The total bottom width is \( 100\text{ cm} \). The right step goes in by \( 40\text{ cm} \). Therefore, the top horizontal side is:

$$ 100\text{ cm} - 40\text{ cm} = 60\text{ cm} $$

• Inner vertical step: The total left height is \( 70\text{ cm} \), and the right height is \( 35\text{ cm} \). The vertical step down is:

$$ 70\text{ cm} - 35\text{ cm} = 35\text{ cm} $$

Now, sum all the sides to find the perimeter:

$$ P_1 = 70 + 100 + 35 + 40 + 60 + 35 = 340\text{ cm} $$

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Step 2: Solve Problem 2

Let's identify all 6 sides of the second shape:

• Right vertical side: \( 115\text{ cm} \)
• Bottom horizontal side: \( 75\text{ cm} \)
• Top horizontal side: \( 36\text{ cm} \)
• Upper left vertical side: \( 45\text{ cm} \)
• Middle horizontal step: The bottom width is \( 75\text{ cm} \) and the top width is \( 36\text{ cm} \). The horizontal step is:

$$ 75\text{ cm} - 36\text{ cm} = 39\text{ cm} $$

• Lower left vertical side: The total right height is \( 115\text{ cm} \) and the upper left vertical side is \( 45\text{ cm} \). The lower left vertical side is:

$$ 115\text{ cm} - 45\text{ cm} = 70\text{ cm} $$

Now, sum all the sides to find the perimeter:

$$ P_2 = 115 + 75 + 36 + 45 + 39 + 70 = 380\text{ cm} $$

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Step 3: Solve Problem 3

Let's identify all 6 sides of the third shape:

• Top horizontal side: \( 125\text{ cm} \)
• Left vertical side: \( 64\text{ cm} \)
• Bottom-left horizontal side: \( 75\text{ cm} \)
• Inner vertical step: \( 32\text{ cm} \)
• Bottom-right horizontal side: The total top width is \( 125\text{ cm} \) and the bottom-left width is \( 75\text{ cm} \). The remaining bottom-right horizontal side is:

$$ 125\text{ cm} - 75\text{ cm} = 50\text{ cm} $$

• Right vertical side: The left height is \( 64\text{ cm} \) and the inner vertical step goes up by \( 32\text{ cm} \). The right vertical side is:

$$ 64\text{ cm} - 32\text{ cm} = 32\text{ cm} $$

Now, sum all the sides to find the perimeter:

$$ P_3 = 125 + 64 + 75 + 32 + 50 + 32 = 378\text{ cm} $$

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Step 4: Solve Problem 4

Let's identify all 6 sides of the fourth shape:

• Top horizontal side: \( 275\text{ cm} \)
• Bottom-right horizontal side: \( 190\text{ cm} \)
• Left vertical side: \( 36\text{ cm} \)
• Inner vertical step: \( 36\text{ cm} \)
• Bottom-left horizontal side: The total top width is \( 275\text{ cm} \) and the bottom-right width is \( 190\text{ cm} \). The bottom-left horizontal side is:

$$ 275\text{ cm} - 190\text{ cm} = 85\text{ cm} $$

• Right vertical side: The left vertical side is \( 36\text{ cm} \) and the step goes down by \( 36\text{ cm} \). The right vertical side is:

$$ 36\text{ cm} + 36\text{ cm} = 72\text{ cm} $$

Now, sum all the sides to find the perimeter:

$$ P_4 = 275 + 190 + 36 + 36 + 85 + 72 = 694\text{ cm} $$

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Answer:

1. \( 340\text{ cm} \)
2. \( 380\text{ cm} \)
3. \( 378\text{ cm} \)
4. \( 694\text{ cm} \)