Question

the upper-left coordinates on a rectangle are (-8, 8), and the upper-right coordinates are (-3, 8). the rectangle has an area of 15 square units. draw the rectangle on the coordinate plane below.

Explanation:

Step1: Find the length of the top side

The upper - left coordinate is \((-8,8)\) and the upper - right coordinate is \((-3,8)\). Since the \(y\) - coordinates are the same, the length of the top side (let's call it \(l\)) is calculated by the difference in the \(x\) - coordinates. Using 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, \(x_1=-8\), \(x_2 = - 3\), so \(l=\vert-3-(-8)\vert=\vert-3 + 8\vert=5\) units.

Step2: Find the height of the rectangle

The area of a rectangle is given by the formula \(A=l\times h\), where \(A\) is the area, \(l\) is the length, and \(h\) is the height. We know that \(A = 15\) square units and \(l = 5\) units. Substituting these values into the formula \(15=5\times h\), we can solve for \(h\) by dividing both sides of the equation by 5: \(h=\frac{15}{5}=3\) units.

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

The upper - left coordinate is \((-8,8)\). To find the lower - left coordinate, we need to move down (decrease the \(y\) - coordinate) by \(h = 3\) units. So the \(y\) - coordinate of the lower - left point is \(8-3 = 5\)? Wait, no, wait. Wait, the upper - left is \((-8,8)\), the upper - right is \((-3,8)\). The height is 3, so the lower - left point will have the same \(x\) - coordinate as the upper - left point \((-8)\) and the \(y\) - coordinate will be \(8 - 3=5\)? Wait, no, maybe I made a mistake. Wait, the distance between the upper and lower sides is the height. Since the upper side is at \(y = 8\), the lower side will be at \(y=8 - 3=5\)? Wait, no, let's re - check. Wait, the area is 15, length is 5, so height is 3. So the vertical distance between the upper and lower sides is 3. So the lower - left point: \(x=-8\), \(y = 8-3 = 5\)? Wait, no, in the coordinate plane, if we go down from \(y = 8\) by 3 units, the \(y\) - coordinate becomes \(8-3 = 5\)? Wait, but in the given graph, there are points at lower \(y\) - values. Wait, maybe I messed up the direction. Wait, maybe the height is the vertical side, so if the upper side is at \(y = 8\), and the height is 3, then the lower side is at \(y=8 - 3=5\)? No, wait, maybe the \(y\) - coordinate decreases as we go down. Wait, no, the standard coordinate system has \(y\) increasing upwards. So if we have a rectangle, the upper side is at \(y = 8\), and the height is 3, so the lower side is at \(y=8 - 3 = 5\)? But in the given graph, there are points with \(y\) - coordinates around 0. Wait, maybe I made a mistake in calculating the length. Wait, the upper - left is \((-8,8)\), upper - right is \((-3,8)\). The distance between \(-8\) and \(-3\) on the \(x\) - axis is \(\vert-3-(-8)\vert=\vert5\vert = 5\), that's correct. Then area is 15, so height is 3. So the lower - left point is \((-8,8 - 3)=(-8,5)\)? No, that doesn't match the graph. Wait, maybe the height is 3, but in the negative \(y\) direction. Wait, maybe the upper side is at \(y = 8\), and the lower side is at \(y=8-3 = 5\)? No, maybe I have the length wrong. Wait, no, the horizontal distance between \((-8,8)\) and \((-3,8)\) is \(|-3-(-8)| = 5\), that's correct. Then area \(A = l\times h\), \(15=5\times h\), so \(h = 3\). So the vertical sides have length 3. So the lower - left corner: \(x=-8\), \(y=8 - 3=5\)? But in the given graph, there are points with \(y\) - coordinates like 0 or - 2. Wait, maybe the problem is that the upper - left is \((-8,8)\), upper - right is \((-3,8)\), so the length is 5, area is 15, so height is 3. So the lower - left is \((-8,8 - 3)=(-8,5)\), lower - right is \((-3,8 - 3)=(-3,5)\). But in the graph provided, maybe there was a mistake in the initial graph, or maybe I misread the coordinates. Anyway, to draw the rectangle:
- The upper side has endpoints \((-8,8)\) and \((-3,8)\).
- The lower side will have endpoints \((-8,8 - 3)=(-8,5)\) and \((-3,8 - 3)=(-3,5)\).
- Then connect the upper - left to lower - left \((-8,8)\) to \((-8,5)\), upper - right to lower - right \((-3,8)\) to \((-3,5)\), and lower - left to lower - right \((-8,5)\) to \((-3,5)\).

(Note: If the graph in the problem has a different scale, maybe the height is calculated differently, but based on the given coordinates and area, the above is the correct way to find the coordinates of the lower vertices.)

Answer:

To draw the rectangle:

1. Plot the upper - left point at \((-8,8)\) and upper - right point at \((-3,8)\).
2. Calculate the length of the horizontal side: \(|-3-(-8)| = 5\) units.
3. Calculate the height: Using \(A = l\times h\), \(15=5\times h\), so \(h = 3\) units.
4. Plot the lower - left point at \((-8,8 - 3)=(-8,5)\) and lower - right point at \((-3,8 - 3)=(-3,5)\).
5. Connect the points: \((-8,8)\) to \((-3,8)\), \((-3,8)\) to \((-3,5)\), \((-3,5)\) to \((-8,5)\), and \((-8,5)\) to \((-8,8)\).

(If the graph's scale is different, adjust the \(y\) - coordinate calculation accordingly, but the process remains the same.)