To determine which triangle has the same area as Quincy's triangle, we first need to find the area of Quincy's triangle. However, we don't have the coordinates of Quincy's triangle provided.
Assuming we know the coordinates of Quincy's triangle, we can use the formula for the area of a triangle given by three vertices (x1, y1), (x2, y2), and (x3, y3):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]
Once we compute the area for Quincy's triangle using its coordinates, we can do the same for each of the triangles provided in the options.
Let's calculate the area for each of the four triangles mentioned:
Triangle One
Vertices: A (2,1) B (7,2) C (2,7) \[ \text{Area} = \frac{1}{2} \left| 2(2-7) + 7(7-1) + 2(1-2) \right| \ = \frac{1}{2} \left| 2(-5) + 7(6) + 2(-1) \right| \ = \frac{1}{2} \left| -10 + 42 - 2 \right| \ = \frac{1}{2} \left| 30 \right| = 15 \]
Triangle Two
Vertices: A (2,1) B (5,9) C (2,7) \[ \text{Area} = \frac{1}{2} \left| 2(9-7) + 5(7-1) + 2(1-9) \right| \ = \frac{1}{2} \left| 2(2) + 5(6) + 2(-8) \right| \ = \frac{1}{2} \left| 4 + 30 - 16 \right| \ = \frac{1}{2} \left| 18 \right| = 9 \]
Triangle Three
Vertices: A (2,1) B (2,7) C (10,2) \[ \text{Area} = \frac{1}{2} \left| 2(7-2) + 2(2-1) + 10(1-7) \right| \ = \frac{1}{2} \left| 2(5) + 2(1) + 10(-6) \right| \ = \frac{1}{2} \left| 10 + 2 - 60 \right| \ = \frac{1}{2} \left| -48 \right| = 24 \]
Triangle Four
Vertices: A (2,1) B (2,7) C (4,4) \[ \text{Area} = \frac{1}{2} \left| 2(7-4) + 2(4-1) + 4(1-7) \right| \ = \frac{1}{2} \left| 2(3) + 2(3) + 4(-6) \right| \ = \frac{1}{2} \left| 6 + 6 - 24 \right| \ = \frac{1}{2} \left| -12 \right| = 6 \]
Summary of Areas
- Triangle One: Area = 15
- Triangle Two: Area = 9
- Triangle Three: Area = 24
- Triangle Four: Area = 6
Comparing the areas, we can say:
- Triangle One has area 15
- Triangle Two has area 9
- Triangle Three has area 24
- Triangle Four has area 6
If we had Quincy's triangle area, we could determine which triangle matches it. Without knowing Quincy's triangle's area, we can't definitively say which option has the same area as Quincy's triangle.
Do you have the area of Quincy's triangle or its vertices? With that, we can easily conclude which one matches.
3 answers