Question

the upper-left coordinates on a rectangle are $(-5, 6)$, and the upper-right coordinates are $(-2, 6)$. the rectangle has a perimeter of 16 units. draw the rectangle on the coordinate plane below.

Explanation:

Step1: Find the length of the top side

The upper - left coordinate is \((-5,6)\) and the upper - right coordinate is \((-2,6)\). Since the \(y\) - coordinates are the same, the length of the top side (let's call it \(l\)) is calculated by the formula for the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) when \(y_1 = y_2\): \(l=\vert x_2 - x_1\vert\). Here, \(x_1=-5\), \(x_2 = - 2\), so \(l=\vert-2-(-5)\vert=\vert-2 + 5\vert=3\) units.

Step2: Use the perimeter formula to find the width

The perimeter of a rectangle is given by the formula \(P = 2(l + w)\), where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width. We know that \(P = 16\) units and \(l = 3\) units. Substitute these values into the formula:
\[16=2(3 + w)\]
First, divide both sides of the equation by 2: \(\frac{16}{2}=3 + w\), so \(8=3 + w\). Then, subtract 3 from both sides: \(w=8 - 3=5\) units.

Step3: Find the lower - left and lower - right coordinates

The upper - left coordinate is \((-5,6)\). To find the lower - left coordinate, we move down (decrease the \(y\) - coordinate) by the width \(w = 5\) units. So the \(y\) - coordinate of the lower - left coordinate is \(6-5 = 1\)? Wait, no, wait. Wait, if the width is 5, and the upper \(y\) - coordinate is 6, then the lower \(y\) - coordinate should be \(6 - 5=1\)? Wait, no, let's check again. Wait, the perimeter formula: \(P = 2(l + w)\), \(l = 3\), \(P = 16\), so \(16=2(3 + w)\Rightarrow8 = 3+w\Rightarrow w = 5\). But the vertical distance from the upper side to the lower side is the width. The upper side is at \(y = 6\), so the lower side should be at \(y=6 - 5 = 1\)? Wait, no, maybe I made a mistake. Wait, the distance between the upper and lower sides is the height (width in vertical direction). Wait, the top side is from \((-5,6)\) to \((-2,6)\), length 3. The perimeter is 16, so \(2(l + w)=16\Rightarrow l + w = 8\Rightarrow w=8 - 3 = 5\). So the vertical change from the upper side to the lower side is 5 units. Since the upper side is at \(y = 6\), the lower side is at \(y=6 - 5=1\)? Wait, but in the graph, there are points at \(y = 2\) and \(y=-2\), maybe my calculation is wrong. Wait, no, let's recalculate. Wait, the distance between \((-5,6)\) and \((-5,y)\) should be the width. Let's use the perimeter formula correctly. Let the length be \(l\) (horizontal side) and the width be \(w\) (vertical side). We found \(l = 3\). Then \(P=2(l + w)\Rightarrow16 = 2(3+w)\Rightarrow8=3 + w\Rightarrow w = 5\). So the vertical distance from the upper horizontal side to the lower horizontal side is 5. So the \(y\) - coordinate of the lower side is \(6-5 = 1\)? But in the given graph, there are points at \(y = 2\) and \(y=-2\), maybe the initial points in the graph are wrong, or maybe I misread the problem. Wait, the problem says "Draw the rectangle on the coordinate plane below". The given graph has points at \((-3,2)\), \((1,2)\), \((-3,-2)\), \((1,-2)\), but that's a different rectangle. Wait, no, the upper - left is \((-5,6)\), upper - right is \((-2,6)\). So the top side is from \(x=-5\) to \(x = - 2\) at \(y = 6\). The length of the top side is \(3\) (since \(-2-(-5)=3\)). The perimeter is 16, so the sum of length and width is 8, so width is \(8 - 3=5\). So the bottom side is at \(y=6 - 5 = 1\)? No, that can't be. Wait, maybe the width is the vertical side, so the distance from \((-5,6)\) to \((-5,6 - w)\) is \(w\). So if \(w = 5\), then the lower - left corner is \((-5,6 - 5)=(-5,1)\), and the lower - right corner is \((-2,6 - 5)=(-2,1)\). Then we can draw the rectangle with vertices at \((-5,6)\), \((-2,6)\), \((-2,1)\), \((-5,1)\).

Answer:

To draw the rectangle:

1. Plot the upper - left vertex at \((-5,6)\) and upper - right vertex at \((-2,6)\).
2. Calculate the width: Using \(P = 2(l + w)\), \(l = 3\) (from \(\vert-2-(-5)\vert\)), \(P = 16\), we get \(w = 5\).
3. Plot the lower - left vertex at \((-5,6 - 5)=(-5,1)\) and lower - right vertex at \((-2,6 - 5)=(-2,1)\).
4. Connect the four vertices \((-5,6)\), \((-2,6)\), \((-2,1)\), \((-5,1)\) to form the rectangle.