Question

what is the perimeter of quadrilateral vwxy? write your answer as an integer or as a decimal rounded to the nearest tenth. units

Explanation:

First, we need to determine the coordinates of each vertex of quadrilateral \(VWXY\). From the graph, we can identify the coordinates:
- \(X(3, 4)\) (assuming the grid lines, let's confirm: looking at the x - axis, it's 3 units from the origin, y - axis 4 units)
- \(W(7, - 5)\)
- \(V(8, - 9)\)
- \(Y(5, - 7)\)

To find the perimeter, we need to calculate the length of each side \(XY\), \(YW\), \(WV\), and \(VX\) (wait, actually the sides are \(XY\), \(YW\), \(WV\), and \(VX\)? Wait, no, the quadrilateral is \(VWXY\), so the sides are \(VW\), \(WX\), \(XY\), and \(YV\)? Wait, let's re - identify the vertices in order. Let's list the coordinates properly:

Looking at the graph:
- \(X\): Let's say \(X(3,4)\) (x = 3, y = 4)
- \(W\): \(W(7, - 5)\) (x = 7, y=-5)
- \(V\): \(V(8, - 9)\) (x = 8, y = - 9)
- \(Y\): \(Y(5, - 7)\) (x = 5, y=-7)

Now, we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the length of each side.

Step 1: Calculate the length of \(WX\)

Coordinates of \(W(7, - 5)\) and \(X(3,4)\)
\(d_{WX}=\sqrt{(3 - 7)^2+(4-(-5))^2}=\sqrt{(- 4)^2+(9)^2}=\sqrt{16 + 81}=\sqrt{97}\approx9.8\)

Step 2: Calculate the length of \(XV\)

Coordinates of \(X(3,4)\) and \(V(8, - 9)\)
\(d_{XV}=\sqrt{(8 - 3)^2+(-9 - 4)^2}=\sqrt{5^2+(-13)^2}=\sqrt{25 + 169}=\sqrt{194}\approx13.9\) Wait, no, maybe I got the order of the vertices wrong. Let's re - order the vertices correctly. Let's assume the quadrilateral is \(V - W - X - Y - V\). So the sides are \(VW\), \(WX\), \(XY\), \(YV\).

Let's re - find the coordinates:
- \(V\): Let's look at the graph, \(V\) is at (8, - 9)
- \(W\): (7, - 5)
- \(X\): (3, 4)
- \(Y\): (5, - 7)

Step 1: Calculate \(VW\)

Coordinates of \(V(8, - 9)\) and \(W(7, - 5)\)
\(d_{VW}=\sqrt{(7 - 8)^2+(-5-(-9))^2}=\sqrt{(-1)^2+(4)^2}=\sqrt{1 + 16}=\sqrt{17}\approx4.1\)

Step 2: Calculate \(WX\)

Coordinates of \(W(7, - 5)\) and \(X(3,4)\)
\(d_{WX}=\sqrt{(3 - 7)^2+(4 - (-5))^2}=\sqrt{(-4)^2+(9)^2}=\sqrt{16 + 81}=\sqrt{97}\approx9.8\)

Step 3: Calculate \(XY\)

Coordinates of \(X(3,4)\) and \(Y(5, - 7)\)
\(d_{XY}=\sqrt{(5 - 3)^2+(-7 - 4)^2}=\sqrt{2^2+(-11)^2}=\sqrt{4 + 121}=\sqrt{125}\approx11.2\)

Step 4: Calculate \(YV\)

Coordinates of \(Y(5, - 7)\) and \(V(8, - 9)\)
\(d_{YV}=\sqrt{(8 - 5)^2+(-9-(-7))^2}=\sqrt{3^2+(-2)^2}=\sqrt{9 + 4}=\sqrt{13}\approx3.6\)

Step 5: Calculate the perimeter

Perimeter \(P=d_{VW}+d_{WX}+d_{XY}+d_{YV}\)
\(P\approx4.1 + 9.8+11.2 + 3.6\)
\(4.1+9.8 = 13.9\); \(13.9+11.2 = 25.1\); \(25.1+3.6 = 28.7\) Wait, maybe my vertex order is wrong. Let's try another approach. Let's list the correct coordinates by looking at the grid:

Looking at the graph:
- \(X\): (3, 4) (x = 3, y = 4)
- \(W\): (7, - 5) (x = 7, y=-5)
- \(V\): (8, - 9) (x = 8, y=-9)
- \(Y\): (5, - 7) (x = 5, y=-7)

Wait, maybe the sides are \(XW\), \(WV\), \(VY\), \(YX\). Let's recalculate:

\(XW\): distance between \(X(3,4)\) and \(W(7, - 5)\): \(\sqrt{(7 - 3)^2+(-5 - 4)^2}=\sqrt{16 + 81}=\sqrt{97}\approx9.8\)

\(WV\): distance between \(W(7, - 5)\) and \(V(8, - 9)\): \(\sqrt{(8 - 7)^2+(-9+5)^2}=\sqrt{1 + 16}=\sqrt{17}\approx4.1\)

\(VY\): distance between \(V(8, - 9)\) and \(Y(5, - 7)\): \(\sqrt{(5 - 8)^2+(-7 + 9)^2}=\sqrt{9 + 4}=\sqrt{13}\approx3.6\)

\(YX\): distance between \(Y(5, - 7)\) and \(X(3,4)\): \(\sqrt{(3 - 5)^2+(4 + 7)^2}=\sqrt{4 + 121}=\sqrt{125}\approx11.2\)

Now sum them up: \(9.8+4.1 + 3.6+11.2=28.7\)

Wait, maybe I made a mistake in coordinates. Let's check the grid again. Let's assume the coordinates are:

- \(X\): (3, 4) (since it's 3 units right on x - axis, 4 up on y - axis)
- \(W\): (7, - 5) (7 right, 5 down)
- \(V\): (8, - 9) (8 right, 9 down)
- \(Y\): (5, - 7) (5 right, 7 down)

Another way: Let's use the distance formula correctly for each side.

Side 1: \(X\) to \(W\): \((x_1,y_1)=(3,4)\), \((x_2,y_2)=(7, - 5)\)
\(d=\sqrt{(7 - 3)^2+(-5 - 4)^2}=\sqrt{16 + 81}=\sqrt{97}\approx9.8\)

Side 2: \(W\) to \(V\): \((x_1,y_1)=(7, - 5)\), \((x_2,y_2)=(8, - 9)\)
\(d=\sqrt{(8 - 7)^2+(-9+5)^2}=\sqrt{1 + 16}=\sqrt{17}\approx4.1\)

Side 3: \(V\) to \(Y\): \((x_1,y_1)=(8, - 9)\), \((x_2,y_2)=(5, - 7)\)
\(d=\sqrt{(5 - 8)^2+(-7 + 9)^2}=\sqrt{9 + 4}=\sqrt{13}\approx3.6\)

Side 4: \(Y\) to \(X\): \((x_1,y_1)=(5, - 7)\), \((x_2,y_2)=(3,4)\)
\(d=\sqrt{(3 - 5)^2+(4 + 7)^2}=\sqrt{4 + 121}=\sqrt{125}\approx11.2\)

Now, sum all sides: \(9.8+4.1+3.6+11.2 = 28.7\)

Answer:

\(28.7\)