Question
What is the area of the triangle?
(-3,3) (-2,4) (6,5)
30 square units
27.5 square units
44 square units
55 square units
(-3,3) (-2,4) (6,5)
30 square units
27.5 square units
44 square units
55 square units
Answers
GPT-4o mini
To find the area of the triangle formed by the points \((-3, 3)\), \((-2, 4)\), and \((6, 5)\), we can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Let's assign the points:
- \((x_1, y_1) = (-3, 3)\)
- \((x_2, y_2) = (-2, 4)\)
- \((x_3, y_3) = (6, 5)\)
Plugging in these points into the formula gives us:
\[
\text{Area} = \frac{1}{2} \left| -3(4 - 5) + (-2)(5 - 3) + 6(3 - 4) \right|
\]
Calculating each term:
1. \( -3(4 - 5) = -3(-1) = 3 \)
2. \( -2(5 - 3) = -2(2) = -4 \)
3. \( 6(3 - 4) = 6(-1) = -6 \)
Now substituting these values back into the area formula:
\[
\text{Area} = \frac{1}{2} \left| 3 - 4 - 6 \right| = \frac{1}{2} \left| -7 \right| = \frac{1}{2} \cdot 7 = 3.5
\]
So the area of the triangle is \(3.5\) square units.
However, it seems there might have been an earlier incorrect calculation by checking the output again.
Here’s a corrected calculation method:
Let’s recalculate more carefully:
Using the area formula again:
\[
\text{Area} = \frac{1}{2} \left| -3(4 - 5) + (-2)(5 - 3) + 6(3 - 4) \right|
\]
Evaluating the components step by step:
\[
= \frac{1}{2} \left| -3(-1) - 2 \times 2 + 6(-1) \right|
\]
\[
= \frac{1}{2} \left| 3 - 4 - 6 \right|
\]
\[
= \frac{1}{2} \left| -7 \right|
\]
\[
= \frac{1}{2} \times 7 = 3.5
\]
Returning to the options:
After checking all calculations seem to lineup, it remains 3.5, thus not matching any choice options likely there is an error in options or calculation interpretations.
The expected output solution aligns with standard area calculation yielding lower 3.5 square units not matching the provided choices of 30, 27.5, 44, and 55.
Therefore the area indeed for the triangle provided instances checks out remains \(3.5\).
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Let's assign the points:
- \((x_1, y_1) = (-3, 3)\)
- \((x_2, y_2) = (-2, 4)\)
- \((x_3, y_3) = (6, 5)\)
Plugging in these points into the formula gives us:
\[
\text{Area} = \frac{1}{2} \left| -3(4 - 5) + (-2)(5 - 3) + 6(3 - 4) \right|
\]
Calculating each term:
1. \( -3(4 - 5) = -3(-1) = 3 \)
2. \( -2(5 - 3) = -2(2) = -4 \)
3. \( 6(3 - 4) = 6(-1) = -6 \)
Now substituting these values back into the area formula:
\[
\text{Area} = \frac{1}{2} \left| 3 - 4 - 6 \right| = \frac{1}{2} \left| -7 \right| = \frac{1}{2} \cdot 7 = 3.5
\]
So the area of the triangle is \(3.5\) square units.
However, it seems there might have been an earlier incorrect calculation by checking the output again.
Here’s a corrected calculation method:
Let’s recalculate more carefully:
Using the area formula again:
\[
\text{Area} = \frac{1}{2} \left| -3(4 - 5) + (-2)(5 - 3) + 6(3 - 4) \right|
\]
Evaluating the components step by step:
\[
= \frac{1}{2} \left| -3(-1) - 2 \times 2 + 6(-1) \right|
\]
\[
= \frac{1}{2} \left| 3 - 4 - 6 \right|
\]
\[
= \frac{1}{2} \left| -7 \right|
\]
\[
= \frac{1}{2} \times 7 = 3.5
\]
Returning to the options:
After checking all calculations seem to lineup, it remains 3.5, thus not matching any choice options likely there is an error in options or calculation interpretations.
The expected output solution aligns with standard area calculation yielding lower 3.5 square units not matching the provided choices of 30, 27.5, 44, and 55.
Therefore the area indeed for the triangle provided instances checks out remains \(3.5\).