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
graph with points r(-7,2), w(1,-3), d(5,7), and (7,7); distance formula box: ( d = sqrt{(quad)^2 + (quad)^2} ), ( d = quad )

Explanation:

Step1: Identify coordinates

Let's list the coordinates of the vertices:
- \( R(-7, 2) \)
- \( W(1, -3) \)
- Let's assume the other two points: Let's say the fourth point is \( Q(7, 7) \) (from the graph, the point with y=7 and x=7) and \( D(5, 7) \) (from the graph, the point with y=7 and x=5)

Step2: Calculate \( RW \) length

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

Step3: Calculate \( WQ \) length

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

Step4: Calculate \( QD \) length

For \( Q(7, 7) \) and \( D(5, 7) \):
\( x_2 - x_1 = 5 - 7 = -2 \)
\( y_2 - y_1 = 7 - 7 = 0 \)
\( QD = \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2 \)

Step5: Calculate \( DR \) length

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

Step6: Calculate perimeter

Perimeter = \( RW + WQ + QD + DR \)
\( \approx 9.43 + 11.66 + 2 + 13 \)
\( = 36.09 \) (Wait, maybe I misidentified the points. Let's re - check the graph. Let's assume the quadrilateral is \( R(-7,2) \), \( W(1, - 3) \), \( Q(1,7) \)? No, the graph has D at (5,7) and Q at (7,7). Wait, maybe the correct points are \( R(-7,2) \), \( W(1, - 3) \), \( Q(7,7) \), \( D(5,7) \). Wait, maybe a better approach: Let's count the grid squares.

Wait, maybe the coordinates are:
- \( R(-7, 2) \)
- \( W(1, -3) \)
- \( Q(7, 7) \)
- \( D(5, 7) \)

Wait, let's recalculate \( DR \): from \( D(5,7) \) to \( R(-7,2) \):
\( \Delta x=-7 - 5=-12 \), \( \Delta y = 2 - 7=-5 \)
\( DR=\sqrt{(-12)^2+(-5)^2}=\sqrt{144 + 25}=\sqrt{169}=13 \) (correct)

\( QD \): from \( Q(7,7) \) to \( D(5,7) \): \( \Delta x = 5 - 7=-2 \), \( \Delta y=0 \), so length is 2 (correct)

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

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

Now sum them up: \( 13+2 + 11.66+9.43=36.09\approx36 \) (maybe there is a miscalculation in point identification. Let's try another way. Maybe the quadrilateral is \( R(-7,2) \), \( W(1, - 3) \), \( Q(1,7) \), \( D(5,7) \). Let's check:

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

\( WQ \): from (1, - 3) to (1,7): vertical distance, \( 7-(-3)=10 \)

\( QD \): from (1,7) to (5,7): horizontal distance, \( 5 - 1 = 4 \)

\( DR \): from (5,7) to (-7,2): \( \sqrt{(-12)^2+(-5)^2}=13 \)

Then perimeter: \( 9.43+10 + 4+13=36.43\approx36 \). Wait, maybe the correct points are \( R(-7,2) \), \( W(1, - 3) \), \( Q(7,7) \), \( D(5,7) \) is wrong. Let's look at the graph again. The point D is at (5,7) and the other point at (7,7) is Q. The point R is at (-7,2) and W is at (1, - 3).

Wait, maybe the distance between D(5,7) and R(-7,2) is 13 (as \( \sqrt{( - 7 - 5)^2+(2 - 7)^2}=\sqrt{(-12)^2+(-5)^2}=13 \)), distance between D(5,7) and Q(7,7) is 2 (horizontal), distance between Q(7,7) and W(1, - 3): \( \sqrt{(1 - 7)^2+(-3 - 7)^2}=\sqrt{(-6)^2+(-10)^2}=\sqrt{36 + 100}=\sqrt{136}\approx11.66 \), distance between W(1, - 3) and R(-7,2): \( \sqrt{( - 7 - 1)^2+(2+3)^2}=\sqrt{(-8)^2+5^2}=\sqrt{64 + 25}=\sqrt{89}\approx9.43 \). Summing these: \( 13 + 2+11.66 + 9.43=36.09\approx36 \).

Alternatively, maybe the grid is 1 unit per square. Let's recalculate with exact values:

\( DR=\sqrt{(-12)^2+(-5)^2}=13 \)

\( QD = 2 \)

\( WQ=\sqrt{6^2 + 10^2}=\sqrt{136}=2\sqrt{34}\approx11.66 \)

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

Perimeter \( = 13+2 + 2\sqrt{34}+\sqrt{89}\approx13 + 2+11.66+9.43 = 36.09\approx36 \) (if we take approximate values) or if we made a mistake in point identification.

Wait, maybe the correct quadrilateral has vertices \( R(-7,2) \), \( W(1, - 3) \), \( Q(1,7) \), \( D(5,7) \). Let's recalculate:

- \( RW \): distance between (-7,2) and (1, - 3): \( \sqrt{(1 + 7)^2+(-3 - 2)^2}=\sqrt{8^2+(-5)^2}=\sqrt{64 + 25}=\sqrt{89}\approx9.43 \)
- \( WQ \): distance between (1, - 3) and (1,7): \( |7 - (-3)| = 10 \)
- \( QD \): distance between (1,7) and (5,7): \( |5 - 1| = 4 \)
- \( DR \): distance between (5,7) and (-7,2): \( \sqrt{(-7 - 5)^2+(2 - 7)^2}=\sqrt{(-12)^2+(-5)^2}=\sqrt{144 + 25}=\sqrt{169}=13 \)

Perimeter \( = 9.43+10 + 4+13=36.43\approx36 \)

Another way: Maybe the graph is such that each square is 1 cm. Let's count the horizontal and vertical distances.

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

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

\( Q(7,7) \) to \( D(5,7) \): horizontal change - 2, vertical change 0, length 2

\( D(5,7) \) to \( R(-7,2) \): horizontal change - 12, vertical change - 5, length 13

Summing these: \( 9.43+11.66 + 2+13=36.09\approx36 \) (or maybe 36 cm)

Answer:

\( 36 \) (approximate, depending on the exact point identification, the perimeter is approximately 36 centimeters)