Question
Daniella is drawing plans for a garden, measured in feet, which is shown below on the coordinate plane. Daniella has two vertices of the garden at points (Negative3, Negative 1) and (4, Negative 1).
On a coordinate plane, points (negative 3, negative 1) and (4, negative 1) are plotted.
At which points should Daniella have the other two vertices in order to make the area of her garden 35 square feet?
(Negative 3, 1) and (4, 1)
(Negative 3, 2) and (4, 2)
(Negative 3, 3) and (4, 3)
(Negative 3, 4) and (4, 4)
On a coordinate plane, points (negative 3, negative 1) and (4, negative 1) are plotted.
At which points should Daniella have the other two vertices in order to make the area of her garden 35 square feet?
(Negative 3, 1) and (4, 1)
(Negative 3, 2) and (4, 2)
(Negative 3, 3) and (4, 3)
(Negative 3, 4) and (4, 4)
Answers
GPT-4o mini
To determine the locations of the other two vertices of the garden in order to achieve an area of 35 square feet, we start by identifying the geometry of the shape being created by the given vertices.
The given vertices \((-3, -1)\) and \((4, -1)\) are horizontally aligned along the line \(y = -1\). This means that the base of the garden is essentially the horizontal segment between these two points.
1. **Calculate the length of the base**:
The length of the base, which runs from \((-3, -1)\) to \((4, -1)\), can be calculated as:
\[
\text{Length} = x_2 - x_1 = 4 - (-3) = 4 + 3 = 7 \text{ feet}
\]
2. **Use the area formula for a rectangle**:
The area \(A\) of a rectangle is given by the formula:
\[
A = \text{base} \times \text{height}
\]
We need the area to be \(35\) square feet, and we already know the base is \(7\) feet. We can let \(h\) be the height of the rectangle (the distance vertically from \(-1\) to the y-coordinates of the other vertices). Therefore, we can set up our equation:
\[
35 = 7 \times h
\]
Solving for \(h\):
\[
h = \frac{35}{7} = 5 \text{ feet}
\]
3. **Determine the positions of the other two vertices**:
The original points have a y-coordinate of \(-1\). To find the other two vertices that are at a height of \(5\) feet above this base, we add \(5\) to \(-1\):
\[
y = -1 + 5 = 4
\]
Thus, the coordinates of the other two vertices would be:
\((-3, 4)\) and \((4, 4)\).
4. **Select the correct option**:
The correct answer for the positions of the other two vertices, in order for Daniella to achieve an area of \(35\) square feet, is **\((-3, 4)\) and \((4, 4)\)**.
So the correct option is:
**(Negative 3, 4) and (4, 4)**.
The given vertices \((-3, -1)\) and \((4, -1)\) are horizontally aligned along the line \(y = -1\). This means that the base of the garden is essentially the horizontal segment between these two points.
1. **Calculate the length of the base**:
The length of the base, which runs from \((-3, -1)\) to \((4, -1)\), can be calculated as:
\[
\text{Length} = x_2 - x_1 = 4 - (-3) = 4 + 3 = 7 \text{ feet}
\]
2. **Use the area formula for a rectangle**:
The area \(A\) of a rectangle is given by the formula:
\[
A = \text{base} \times \text{height}
\]
We need the area to be \(35\) square feet, and we already know the base is \(7\) feet. We can let \(h\) be the height of the rectangle (the distance vertically from \(-1\) to the y-coordinates of the other vertices). Therefore, we can set up our equation:
\[
35 = 7 \times h
\]
Solving for \(h\):
\[
h = \frac{35}{7} = 5 \text{ feet}
\]
3. **Determine the positions of the other two vertices**:
The original points have a y-coordinate of \(-1\). To find the other two vertices that are at a height of \(5\) feet above this base, we add \(5\) to \(-1\):
\[
y = -1 + 5 = 4
\]
Thus, the coordinates of the other two vertices would be:
\((-3, 4)\) and \((4, 4)\).
4. **Select the correct option**:
The correct answer for the positions of the other two vertices, in order for Daniella to achieve an area of \(35\) square feet, is **\((-3, 4)\) and \((4, 4)\)**.
So the correct option is:
**(Negative 3, 4) and (4, 4)**.