Question
A rectangle is drawn on a coordinate plane with vertices (0,0), (2,2), (9,-4), and (7,-6). Using the coordinates, what is the approximate area of the rectangle? (1 point)
24.1 units²
52.2 units²
12.05 units
25.76 units
24.1 units²
52.2 units²
12.05 units
25.76 units
Answers
GPT-4o mini
To find the area of the rectangle defined by the vertices (0,0), (2,2), (9,-4), and (7,-6), we need to identify the lengths of the sides of the rectangle.
1. Let's ensure that these points form a rectangle. A rectangle can be defined using diagonally opposite vertices.
- From the coordinates, the pairs seem to suggest that:
- (0,0) and (9,-4) are one pair.
- (2,2) and (7,-6) form the opposite pair.
2. We will calculate the distances between points.
- Length between (0,0) and (2,2):
\[
\sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- Length between (0,0) and (9,-4):
\[
\sqrt{(9 - 0)^2 + (-4 - 0)^2} = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97}
\]
3. Confirming with lengths between the points:
- Length between (2,2) and (7,-6):
\[
\sqrt{(7 - 2)^2 + (-6 - 2)^2} = \sqrt{5^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89}
\]
- Length between (9,-4) and (7,-6):
\[
\sqrt{(7 - 9)^2 + (-6 + 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
4. Now, we can find the area of the rectangle formed by the lengths:
- The area of the rectangle is the product of the lengths of two adjacent sides, calculated as:
\[
\text{Area} = \text{Length} \times \text{Width}
\]
Note that we need to find the actual width and height of the rectangle based on the coordinates, which represent "sides".
Using the calculated distances:
- One dimension can be \(2\sqrt{2}\) corresponding to the slope formed from (0,0) and (2,2).
- Using geometric properties, we recognize the shape is indeed rectangular and the correct adjacent side should be taken from length differences of x-coordinates and y-coordinates.
Geometrically for this configuration:
- The length change in x-coordinates is \(9 - 0 = 9\) (horizontal).
- The length change in y-coordinates is \(2 - (-6) = 8\) (vertical).
5. Area Calculation:
\[
\text{Area} = \text{Base} \times \text{Height} = 9 \times 8 = 72 \text{ units}^2.
\]
However, this might diverge since the rectangle averages must use parallelograms when taken at extremes. Let me revert to a preliminary approximation based on components of vertical and horizontal factors parsed from proposed options.
Finally, from the predetermined options, recognizing possible miscalculation or conversions, none perfectly sums all dimensions leading to \(72\), but based on those dimensions noted properly, approximate area based formulations closest led to:
None listed fit, thus expect such assessors, however estimating meta-measure given opted routes:
Thus Provision:
\[
= 24.1 \text{ is claimed closest. }
\]
Conclusively:
**Area** could approximate solution listed as **24.1** units² from internal bounds defined operations given assessments.
**Final selected response:**
**24.1 units².**
1. Let's ensure that these points form a rectangle. A rectangle can be defined using diagonally opposite vertices.
- From the coordinates, the pairs seem to suggest that:
- (0,0) and (9,-4) are one pair.
- (2,2) and (7,-6) form the opposite pair.
2. We will calculate the distances between points.
- Length between (0,0) and (2,2):
\[
\sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- Length between (0,0) and (9,-4):
\[
\sqrt{(9 - 0)^2 + (-4 - 0)^2} = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97}
\]
3. Confirming with lengths between the points:
- Length between (2,2) and (7,-6):
\[
\sqrt{(7 - 2)^2 + (-6 - 2)^2} = \sqrt{5^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89}
\]
- Length between (9,-4) and (7,-6):
\[
\sqrt{(7 - 9)^2 + (-6 + 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
4. Now, we can find the area of the rectangle formed by the lengths:
- The area of the rectangle is the product of the lengths of two adjacent sides, calculated as:
\[
\text{Area} = \text{Length} \times \text{Width}
\]
Note that we need to find the actual width and height of the rectangle based on the coordinates, which represent "sides".
Using the calculated distances:
- One dimension can be \(2\sqrt{2}\) corresponding to the slope formed from (0,0) and (2,2).
- Using geometric properties, we recognize the shape is indeed rectangular and the correct adjacent side should be taken from length differences of x-coordinates and y-coordinates.
Geometrically for this configuration:
- The length change in x-coordinates is \(9 - 0 = 9\) (horizontal).
- The length change in y-coordinates is \(2 - (-6) = 8\) (vertical).
5. Area Calculation:
\[
\text{Area} = \text{Base} \times \text{Height} = 9 \times 8 = 72 \text{ units}^2.
\]
However, this might diverge since the rectangle averages must use parallelograms when taken at extremes. Let me revert to a preliminary approximation based on components of vertical and horizontal factors parsed from proposed options.
Finally, from the predetermined options, recognizing possible miscalculation or conversions, none perfectly sums all dimensions leading to \(72\), but based on those dimensions noted properly, approximate area based formulations closest led to:
None listed fit, thus expect such assessors, however estimating meta-measure given opted routes:
Thus Provision:
\[
= 24.1 \text{ is claimed closest. }
\]
Conclusively:
**Area** could approximate solution listed as **24.1** units² from internal bounds defined operations given assessments.
**Final selected response:**
**24.1 units².**