Question

samantha is cutting fabric pieces to make patches for her jacket. in order to know how much thread will be required for the stitching, she first needs to know the perimeter of the piece. calculate the perimeter of the piece as shown. enter the unknown segment length(s), then calculate the perimeter of the polygon. perimeter of the quadrilateral: centimeters

Explanation:

Step1: Find length of R to W

Using distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), for \( R(-7, 2) \) and \( W(1, -3) \):
\( x_2 - x_1 = 1 - (-7) = 8 \), \( y_2 - y_1 = -3 - 2 = -5 \)
\( d_{RW} = \sqrt{8^2 + (-5)^2} = \sqrt{64 + 25} = \sqrt{89} \approx 9.43 \)

Step2: Find length of W to Q (assuming Q is (7,7), typo in D(5,88,7) should be Q(7,7))

For \( W(1, -3) \) and \( Q(7, 7) \):
\( x_2 - x_1 = 7 - 1 = 6 \), \( y_2 - y_1 = 7 - (-3) = 10 \)
\( d_{WQ} = \sqrt{6^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136} \approx 11.66 \)

Step3: Find length of Q to D (D(5,7))

For \( Q(7, 7) \) and \( D(5, 7) \):
\( d_{QD} = |7 - 5| = 2 \) (horizontal line, y same)

Step4: Find length of D to R

For \( D(5, 7) \) and \( R(-7, 2) \):
\( x_2 - x_1 = -7 - 5 = -12 \), \( y_2 - y_1 = 2 - 7 = -5 \)
\( d_{DR} = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \)

Step5: Calculate perimeter

Perimeter = \( d_{RW} + d_{WQ} + d_{QD} + d_{DR} \)
\( \approx 9.43 + 11.66 + 2 + 13 = 36.09 \) (Wait, correction: Let's re - check coordinates. Assume correct points: R(-7,2), W(1,-3), D(5,7), Q(7,7)

Wait, recalculate each distance:

1. \( RW \): \( \sqrt{(1 - (-7))^2 + (-3 - 2)^2}=\sqrt{8^2+(-5)^2}=\sqrt{64 + 25}=\sqrt{89}\approx9.43 \)

2. \( WQ \): \( \sqrt{(7 - 1)^2+(7 - (-3))^2}=\sqrt{6^2 + 10^2}=\sqrt{36+100}=\sqrt{136}\approx11.66 \)

3. \( QD \): \( |7 - 5| = 2 \) (since y - coordinates are both 7, horizontal distance)

4. \( DR \): \( \sqrt{(-7 - 5)^2+(2 - 7)^2}=\sqrt{(-12)^2+(-5)^2}=\sqrt{144 + 25}=\sqrt{169}=13 \)

Now sum: \( 9.43+11.66 + 2+13=36.09 \). But maybe the typo in D(5,88,7) is D(5,7). Alternatively, if D is (5,7) and Q is (7,7), then:

Wait, another way: Let's list all sides:

- \( R(-7,2) \) to \( W(1,-3) \): \( \sqrt{(1 + 7)^2+(-3 - 2)^2}=\sqrt{64 + 25}=\sqrt{89}\approx9.43 \)

- \( W(1,-3) \) to \( D(5,7) \): \( \sqrt{(5 - 1)^2+(7 + 3)^2}=\sqrt{16 + 100}=\sqrt{116}\approx10.77 \) (Wait, I think I misidentified Q. Maybe the quadrilateral is R, W, D, and another point? Wait the graph has R(-7,2), W(1,-3), D(5,7), and (7,7). Let's check the coordinates again.

Wait the point D is (5,7) and the other point is (7,7). So the quadrilateral is R(-7,2), W(1,-3), (7,7), D(5,7)? No, the blue lines: R to W, W to (7,7)? Wait the blue line from W(1,-3) goes to (7,7)? Wait the user's graph: R(-7,2), W(1,-3), D(5,7), and (7,7). Let's use correct distance formula for each side:

1. \( R(-7,2) \) to \( W(1,-3) \):
\( \Delta x = 1 - (-7)=8 \), \( \Delta y=-3 - 2=-5 \)
\( d_1=\sqrt{8^2+(-5)^2}=\sqrt{64 + 25}=\sqrt{89}\approx9.43 \)

2. \( W(1,-3) \) to \( (7,7) \):
\( \Delta x = 7 - 1 = 6 \), \( \Delta y=7 - (-3)=10 \)
\( d_2=\sqrt{6^2 + 10^2}=\sqrt{36+100}=\sqrt{136}\approx11.66 \)

3. \( (7,7) \) to \( D(5,7) \):
\( \Delta x = 5 - 7=-2 \), \( \Delta y=7 - 7 = 0 \)
\( d_3=\sqrt{(-2)^2+0^2}=2 \)

4. \( D(5,7) \) to \( R(-7,2) \):
\( \Delta x=-7 - 5=-12 \), \( \Delta y=2 - 7=-5 \)
\( d_4=\sqrt{(-12)^2+(-5)^2}=\sqrt{144 + 25}=\sqrt{169}=13 \)

Now sum all sides: \( 9.43+11.66 + 2+13 = 36.09 \approx36.1 \). But maybe there is a miscalculation. Wait, let's check \( W(1,-3) \) to \( D(5,7) \) directly (if the quadrilateral is R, W, D, and (7,7) is a typo). If the quadrilateral is R(-7,2), W(1,-3), D(5,7), and back to R, then:

\( RW=\sqrt{8^2+(-5)^2}=\sqrt{89}\approx9.43 \)

\( WD=\sqrt{(5 - 1)^2+(7 + 3)^2}=\sqrt{16 + 100}=\sqrt{116}\approx10.77 \)

\( DR=\sqrt{(-7 - 5)^2+(2 - 7)^2}=\sqrt{144 + 25}=13 \)

And the fourth side: Wait, maybe the fourth point is (7,7), so the quadrilateral is R, W, (7,7), D. Then the sides are RW, W(7,7), (7,7)D, DR.

Alternatively, maybe the grid is 1 unit per square. Let's count the horizontal and vertical distances.

For \( R(-7,2) \) to \( W(1,-3) \): horizontal distance 8 (from x=-7 to x=1), vertical distance 5 (from y=2 to y=-3), so length \( \sqrt{8^2 + 5^2}=\sqrt{89}\approx9.43 \)

For \( W(1,-3) \) to \( (7,7) \): horizontal distance 6 (x=1 to x=7), vertical distance 10 (y=-3 to y=7), length \( \sqrt{6^2+10^2}=\sqrt{136}\approx11.66 \)

For \( (7,7) \) to \( D(5,7) \): horizontal distance 2 (x=7 to x=5), vertical distance 0, length 2

For \( D(5,7) \) to \( R(-7,2) \): horizontal distance 12 (x=5 to x=-7), vertical distance 5 (y=7 to y=2), length \( \sqrt{12^2 + 5^2}=13 \)

Summing these: \( 9.43+11.66 + 2+13 = 36.09 \approx36 \) (rounded)

Wait, maybe the problem has integer lengths. Let's re - check the coordinates. Maybe the point D is (5,7) and the other point is (7,7), and the quadrilateral is a polygon with sides:

- \( R(-7,2) \) to \( W(1,-3) \): \( \sqrt{(1 + 7)^2+(-3 - 2)^2}=\sqrt{64 + 25}=\sqrt{89}\approx9.43 \)

- \( W(1,-3) \) to \( (7,7) \): \( \sqrt{(7 - 1)^2+(7 + 3)^2}=\sqrt{36 + 100}=\sqrt{136}\approx11.66 \)

- \( (7,7) \) to \( D(5,7) \): 2 (horizontal)

- \( D(5,7) \) to \( R(-7,2) \): \( \sqrt{(-7 - 5)^2+(2 - 7)^2}=\sqrt{144 + 25}=13 \)

Total perimeter: \( 9.43+11.66+2 + 13=36.09\approx36 \) (or more accurately, if we use exact values: \( \sqrt{89}+\sqrt{136}+2 + 13 \). But maybe there is a mistake in my coordinate identification.

Wait, another approach: Let's assume the quadrilateral has vertices R(-7,2), W(1,-3), D(5,7), and Q(7,7). Let's calculate each side:

1. \( RW \): distance between (-7,2) and (1,-3): \( \sqrt{(1 + 7)^2+(-3 - 2)^2}=\sqrt{64 + 25}=\sqrt{89}\approx9.43 \)

2. \( WQ \): distance between (1,-3) and (7,7): \( \sqrt{(7 - 1)^2+(7 + 3)^2}=\sqrt{36 + 100}=\sqrt{136}\approx11.66 \)

3. \( QD \): distance between (7,7) and (5,7): \( \sqrt{(5 - 7)^2+(7 - 7)^2}=\sqrt{4 + 0}=2 \)

4. \( DR \): distance between (5,7) and (-7,2): \( \sqrt{(-7 - 5)^2+(2 - 7)^2}=\sqrt{144 + 25}=\sqrt{169}=13 \)

Now sum all these lengths: \( \sqrt{89}+\sqrt{136}+2 + 13\approx9.43 + 11.66+2 + 13 = 36.09 \). So the perimeter is approximately 36.1 centimeters (or 36 if rounded to the nearest whole number).

Answer:

\( \approx 36.1 \) (or 36, depending on rounding)