Question

what is the perimeter of rectangle vwxy? write your answer as an integer or as a decimal rounded to the ne perimeter = \boxed{} units

Explanation:

Step1: Identify coordinates of vertices

First, we find the coordinates of each vertex of the rectangle \(VWXY\):
- \(V\): Let's assume from the grid, \(V = (-3, -3)\) (wait, looking at the grid, let's re - check. Wait, the grid lines: Let's find the coordinates properly.
Looking at the graph:
- \(W\): The x - coordinate is \(-9\), y - coordinate is \(5\)? Wait, no, looking at the grid, the points:
Wait, \(W\) is at \((-9,5)\)? Wait, no, the grid has x from - 10 to 10 and y from - 10 to 10. Let's look at the points:
- \(W\): Let's see, the x - coordinate is \(-9\), y - coordinate is \(5\)? Wait, no, the blue dot for \(W\) is at \(x=-9\), \(y = 5\)? Wait, no, the y - axis: the horizontal lines are y - values. Wait, the point \(W\): x=-9, y = 5? Wait, no, the point \(X\) is at \(x=-5\), \(y = 8\)? Wait, maybe a better way: use the distance formula between two points.

Alternatively, since it's a rectangle, opposite sides are equal. Let's find the length and width using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

First, find coordinates:
- \(W(-9,5)\), \(X(-5,8)\), \(V(-3,-3)\), \(Y(1,0)\)

Wait, let's check the distance between \(W\) and \(X\):
\(x_1=-9,y_1 = 5\); \(x_2=-5,y_2 = 8\)
\(d_{WX}=\sqrt{(-5+9)^2+(8 - 5)^2}=\sqrt{4^2+3^2}=\sqrt{16 + 9}=\sqrt{25}=5\)

Distance between \(X\) and \(Y\):
\(x_1=-5,y_1 = 8\); \(x_2=1,y_2 = 0\)
\(d_{XY}=\sqrt{(1 + 5)^2+(0 - 8)^2}=\sqrt{6^2+(-8)^2}=\sqrt{36 + 64}=\sqrt{100}=10\)

Since it's a rectangle, perimeter \(P = 2\times(\text{length}+\text{width})\)
So length \(l = 10\), width \(w = 5\)

Step2: Calculate the perimeter

Perimeter of a rectangle is given by the formula \(P=2(l + w)\)
Substitute \(l = 10\) and \(w = 5\) into the formula:
\(P=2(10 + 5)=2\times15 = 30\)? Wait, no, wait, maybe I made a mistake in coordinates.

Wait, let's re - identify the coordinates correctly. Let's look at the grid:

- Point \(W\): x=-9, y = 5? Wait, no, the x - axis: the vertical lines are x - values. Let's count the grid squares.

Looking at the graph:

- \(W\) is at \((-9,5)\)? Wait, no, the point \(W\): from the origin (0,0), moving left 9 units (x=-9) and up 5 units (y = 5).

- \(X\): left 5 units (x=-5) and up 8 units (y = 8).

- \(V\): left 3 units (x=-3) and down 3 units (y=-3).

- \(Y\): right 1 unit (x = 1) and y = 0.

Wait, maybe another pair of sides. Let's find the distance between \(W\) and \(V\):

\(W(-9,5)\), \(V(-3,-3)\)
\(d_{WV}=\sqrt{(-3 + 9)^2+(-3 - 5)^2}=\sqrt{6^2+(-8)^2}=\sqrt{36 + 64}=\sqrt{100}=10\)

Distance between \(V\) and \(Y\):

\(V(-3,-3)\), \(Y(1,0)\)
\(d_{VY}=\sqrt{(1 + 3)^2+(0 + 3)^2}=\sqrt{4^2+3^2}=\sqrt{16 + 9}=\sqrt{25}=5\)

Ah, so the length of the rectangle is \(10\) (distance between \(W\) and \(V\)) and the width is \(5\) (distance between \(V\) and \(Y\)).

Then the perimeter of a rectangle is \(P = 2\times(\text{length}+\text{width})\)
So \(P=2\times(10 + 5)=2\times15 = 30\)? Wait, no, wait, in a rectangle, opposite sides are equal. So if \(WV = 10\) and \(VY = 5\), then the perimeter is \(2\times(10 + 5)=30\)? Wait, but let's check with another pair.

Distance between \(W\) and \(X\): \(W(-9,5)\), \(X(-5,8)\)
\(d=\sqrt{(-5 + 9)^2+(8 - 5)^2}=\sqrt{16 + 9}=5\)

Distance between \(X\) and \(Y\): \(X(-5,8)\), \(Y(1,0)\)
\(d=\sqrt{(1 + 5)^2+(0 - 8)^2}=\sqrt{36 + 64}=10\)

Yes, so the length is \(10\) and the width is \(5\). Then the perimeter \(P=2\times(10 + 5)=30\)? Wait, no, wait, maybe I messed up the length and width. Wait, no, in a rectangle, perimeter is \(2(l + w)\), where \(l\) is length and \(w\) is width. So if \(l = 10\) and \(w = 5\), then \(P = 2(10 + 5)=30\). But wait, let's check the coordinates again.

Wait, maybe the coordinates are:

- \(W(-9,5)\)

- \(X(-5,8)\)

- \(V(-3,-3)\)

- \(Y(1,0)\)

Wait, let's use the distance formula between \(W\) and \(Y\)? No, better to use the fact that in a rectangle, the length and width can be found by the horizontal and vertical distances.

Wait, another approach: count the number of grid units.

Looking at the horizontal and vertical changes.

From \(W(-9,5)\) to \(X(-5,8)\): the horizontal change is \(-5-(-9)=4\), vertical change is \(8 - 5 = 3\), so the distance is \(5\) (since \(3 - 4-5\) triangle).

From \(X(-5,8)\) to \(Y(1,0)\): horizontal change is \(1-(-5)=6\), vertical change is \(0 - 8=-8\), distance is \(10\) (6 - 8 - 10 triangle).

So the two adjacent sides are \(5\) and \(10\). Then the perimeter is \(2\times(5 + 10)=30\).

Wait, but let's confirm with the coordinates of \(V\) and \(W\):

From \(V(-3,-3)\) to \(W(-9,5)\): horizontal change \(-9-(-3)=-6\), vertical change \(5-(-3)=8\), distance is \(\sqrt{(-6)^2+8^2}=\sqrt{36 + 64}=10\).

From \(V(-3,-3)\) to \(Y(1,0)\): horizontal change \(1-(-3)=4\), vertical change \(0-(-3)=3\), distance is \(\sqrt{4^2+3^2}=5\).

Yes, so the sides are \(10\) and \(5\). So the perimeter is \(2\times(10 + 5)=30\).

Wait, but maybe I made a mistake. Let's check again.

Wait, the formula for the perimeter of a rectangle is \(P = 2(l + w)\), where \(l\) is the length of one side and \(w\) is the length of the adjacent side.

We found that one side (e.g., \(WV\)) has length \(10\) and the adjacent side (e.g., \(VY\)) has length \(5\). So \(P=2\times(10 + 5)=30\).

Answer:

30