Question

the perimeter of the figure below is 71 in. find the length of the missing side.
(note: diagram is not to scale)
answer attempt 1 out of 2

Explanation:

Step1: Sum known sides

First, identify and sum all the known side lengths. The known sides are: \( 9.1 \), \( 6.2 \), \( 6.2 \), \( 6.2 \), \( 10.1 \), \( 3.4 \), \( 9.1 \), \( 3.4 \), \( 6.2 \). Wait, actually, let's list them properly. Looking at the figure: top horizontal \( 9.1 \), two vertical \( 6.2 \) (left and right top), two horizontal \( 6.2 \) (left and right middle top), left slant \( 10.1 \), left vertical bottom \( 3.4 \), bottom horizontal \( 9.1 \), right vertical bottom \( 3.4 \), and the right slant is missing (let's call it \( x \)). Wait, actually, let's count all sides:

Top: \( 9.1 \)

Left top vertical: \( 6.2 \)

Left middle horizontal: \( 6.2 \)

Left slant: \( 10.1 \)

Left bottom vertical: \( 3.4 \)

Bottom: \( 9.1 \)

Right bottom vertical: \( 3.4 \)

Right middle horizontal: \( 6.2 \)

Right top vertical: \( 6.2 \)

Right slant: \( x \)

Now sum the known ones:

\( 9.1 + 6.2 + 6.2 + 10.1 + 3.4 + 9.1 + 3.4 + 6.2 + 6.2 \)

Let's calculate step by step:

\( 9.1 + 9.1 = 18.2 \)

\( 6.2 + 6.2 + 6.2 + 6.2 = 6.2 \times 4 = 24.8 \)

\( 10.1 + 3.4 + 3.4 = 10.1 + 6.8 = 16.9 \)

Now total known: \( 18.2 + 24.8 + 16.9 = 59.9 \)

Step2: Subtract from perimeter

The perimeter is \( 71 \) in. So the missing side \( x = 71 - 59.9 = 11.1 \)? Wait, no, wait, maybe I miscounted the sides. Wait, let's re-express the figure. Maybe it's a symmetric figure? Wait, the left slant is \( 10.1 \), so the right slant should be equal? Wait, no, the perimeter is the sum of all outer sides. Wait, maybe I made a mistake in counting. Let's list all outer sides:

1. Top horizontal: \( 9.1 \)
2. Right top vertical: \( 6.2 \)
3. Right middle horizontal: \( 6.2 \)
4. Right slant: \( x \) (missing)
5. Right bottom vertical: \( 3.4 \)
6. Bottom horizontal: \( 9.1 \)
7. Left bottom vertical: \( 3.4 \)
8. Left slant: \( 10.1 \)
9. Left middle horizontal: \( 6.2 \)
10. Left top vertical: \( 6.2 \)

Wait, that's 10 sides? Wait, no, maybe the figure has 10 sides? Wait, no, let's count again. Maybe the top, two verticals, two horizontals, two slants, two verticals, bottom. Wait, maybe I overcounted. Let's use the perimeter formula: perimeter is sum of all sides. Let's list each side with their lengths:

- Top: \( 9.1 \)
- Right top vertical: \( 6.2 \)
- Right middle horizontal: \( 6.2 \)
- Right slant: \( x \) (unknown)
- Right bottom vertical: \( 3.4 \)
- Bottom: \( 9.1 \)
- Left bottom vertical: \( 3.4 \)
- Left slant: \( 10.1 \)
- Left middle horizontal: \( 6.2 \)
- Left top vertical: \( 6.2 \)

Now sum the known ones: \( 9.1 + 6.2 + 6.2 + 3.4 + 9.1 + 3.4 + 10.1 + 6.2 + 6.2 \)

Let's calculate that:

First, \( 9.1 + 9.1 = 18.2 \)

\( 6.2 + 6.2 + 6.2 + 6.2 = 6.2 \times 4 = 24.8 \)

\( 3.4 + 3.4 = 6.8 \)

\( 10.1 \)

Now sum these: \( 18.2 + 24.8 = 43 \); \( 43 + 6.8 = 49.8 \); \( 49.8 + 10.1 = 59.9 \)

So total known sum is \( 59.9 \). The perimeter is \( 71 \), so:

Step3: Solve for x

Perimeter = sum of all sides, so \( 59.9 + x = 71 \)

Subtract \( 59.9 \) from both sides: \( x = 71 - 59.9 = 11.1 \)

Wait, but wait, maybe I missed a side? Wait, no, let's check again. Wait, the left slant is \( 10.1 \), so the right slant should be equal? Wait, no, maybe the figure is symmetric? Wait, no, the left slant is \( 10.1 \), so maybe the right slant is also \( 10.1 \)? But no, the perimeter is 71. Wait, maybe my initial counting is wrong. Let's try again.

Wait, maybe the sides are:

Top: \( 9.1 \)

Right top vertical: \( 6.2 \)

Right middle horizontal: \( 6.2 \)

Right slant: \( x \)

Right bottom vertical: \( 3.4 \)

Bottom: \( 9.1 \)

Left bottom vertical: \( 3.4 \)

Left slant: \( 10.1 \)

Left middle horizontal: \( 6.2 \)

Left top vertical: \( 6.2 \)

Wait, that's 10 sides. So sum of known: \( 9.1 + 6.2 + 6.2 + 3.4 + 9.1 + 3.4 + 10.1 + 6.2 + 6.2 \)

Wait, that's 9 sides? Wait, no, the right slant is the 10th. So known sides: 9 sides? Wait, no, the right slant is unknown, so the other 9 sides are known? Wait, no, let's count the number of sides:

- Top (1)
- Right top vertical (2)
- Right middle horizontal (3)
- Right slant (4)
- Right bottom vertical (5)
- Bottom (6)
- Left bottom vertical (7)
- Left slant (8)
- Left middle horizontal (9)
- Left top vertical (10)

Yes, 10 sides. So 9 known, 1 unknown. Wait, no, the left slant is 10.1 (known), right slant is x (unknown). So known sides: 9.1, 6.2, 6.2, 3.4, 9.1, 3.4, 10.1, 6.2, 6.2. Wait, that's 9 sides? Wait, no, 9.1 (1), 6.2 (2), 6.2 (3), 3.4 (4), 9.1 (5), 3.4 (6), 10.1 (7), 6.2 (8), 6.2 (9). Then the 10th is x. So sum of known: 9.1 + 6.2 + 6.2 + 3.4 + 9.1 + 3.4 + 10.1 + 6.2 + 6.2. Let's calculate:

9.1 + 9.1 = 18.2

6.2 + 6.2 + 6.2 + 6.2 = 24.8 (four 6.2s)

3.4 + 3.4 = 6.8

10.1

Now sum: 18.2 + 24.8 = 43; 43 + 6.8 = 49.8; 49.8 + 10.1 = 59.9

So total known sum: 59.9. Perimeter is 71, so x = 71 - 59.9 = 11.1? Wait, but that seems high. Wait, maybe I missed a side. Wait, maybe the top, two verticals, two horizontals, two slants, two verticals, bottom. Wait, maybe the figure is made of a rectangle with extensions. Wait, alternatively, maybe the left slant and right slant are equal? But the left is 10.1, so maybe right is also 10.1? But then perimeter would be 59.9 + 10.1 = 70, which is less than 71. So that's not it. Wait, maybe my initial side count is wrong. Let's try another approach. Let's list all sides with their lengths:

- Top: 9.1
- Right top vertical: 6.2
- Right middle horizontal: 6.2
- Right slant: x
- Right bottom vertical: 3.4
- Bottom: 9.1
- Left bottom vertical: 3.4
- Left slant: 10.1
- Left middle horizontal: 6.2
- Left top vertical: 6.2

Wait, that's 10 sides. Now sum all known: 9.1 + 6.2 + 6.2 + 3.4 + 9.1 + 3.4 + 10.1 + 6.2 + 6.2. Wait, that's 9 sides (missing the right slant). So sum:

9.1 (top) + 6.2 (right top vertical) + 6.2 (right middle horizontal) + 3.4 (right bottom vertical) + 9.1 (bottom) + 3.4 (left bottom vertical) + 10.1 (left slant) + 6.2 (left middle horizontal) + 6.2 (left top vertical) = let's calculate:

9.1 + 9.1 = 18.2

6.2 + 6.2 + 6.2 + 6.2 = 24.8 (four 6.2s)

3.4 + 3.4 = 6.8

10.1

Now 18.2 + 24.8 = 43; 43 + 6.8 = 49.8; 49.8 + 10.1 = 59.9

So total known: 59.9. Perimeter is 71, so x = 71 - 59.9 = 11.1? Wait, but that seems odd. Wait, maybe I made a mistake in the number of 6.2s. Let's check the figure again. The left has a 6.2 (top vertical), 6.2 (middle horizontal). The right has 6.2 (top vertical), 6.2 (middle horizontal). So that's four 6.2s? Wait, left top vertical: 6.2, left middle horizontal: 6.2. Right top vertical: 6.2, right middle horizontal: 6.2. So four 6.2s. Then left slant: 10.1, right slant: x. Left bottom vertical: 3.4, right bottom vertical: 3.4. Top: 9.1, bottom: 9.1. So total sides:

Top (9.1) + right top vertical (6.2) + right middle horizontal (6.2) + right slant (x) + right bottom vertical (3.4) + bottom (9.1) + left bottom vertical (3.4) + left slant (10.1) + left middle horizontal (6.2) + left top vertical (6.2). Yes, that's 10 sides. So sum of known: 9.1 + 6.2 + 6.2 + 3.4 + 9.1 + 3.4 + 10.1 + 6.2 + 6.2 = 59.9. Then x = 71 - 59.9 = 11.1. Wait, but let's check again. Maybe the left slant and right slant are both 10.1? But then 59.9 + 10.1 = 70, which is less than 71. So no. Alternatively, maybe I missed a side. Wait, maybe the top and bottom are both 9.1, so two 9.1s. The verticals: two 6.2s (top) and two 3.4s (bottom), so total verticals: 6.2*2 + 3.4*2 = 12.4 + 6.8 = 19.2. The horizontals: 6.2*2 (middle) + 9.1*2 (top and bottom) = 12.4 + 18.2 = 30.6. The slants: 10.1 + x. So total perimeter: 19.2 + 30.6 + 10.1 + x = 59.9 + x. Set equal to 71: 59.9 + x = 71 → x = 71 - 59.9 = 11.1. So that's the calculation.

Answer:

The length of the missing side is \( \boldsymbol{11.1} \) inches.