Bot GPT 4-o mini To find the missing side lengths, perimeter, and area of triangle ABC on the coordinate plane, we'll need the specific coordinates for points A, B, and C. However, since you haven't provided the coordinates, I will demonstrate the procedure using example coordinates. You can use this method with your actual triangle's coordinates for the requested information.
Step 1: Determine the Coordinates
Let's say the coordinates of triangle ABC are:
- A(1, 2)
- B(4, 6)
- C(7, 2)
Step 2: Calculate Side Lengths
-
AB: Using the distance formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\): \[ AB = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
-
BC: \[ BC = \sqrt{(7 - 4)^2 + (2 - 6)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
-
AC: \[ AC = \sqrt{(7 - 1)^2 + (2 - 2)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6 \]
Step 3: Calculate the Perimeter
The perimeter is the sum of the side lengths: \[ \text{Perimeter} = AB + BC + AC = 5 + 5 + 6 = 16 \]
Step 4: Calculate the Area
For the area of triangle ABC, we can first find the height (h) from point A perpendicular to base BC. Since points B and C are on y=2, the height can be calculated as follows:
- The vertical distance from A (1, 2) to line BC will be the y-coordinate difference since A is already on the same y-coordinate as BC.
Using BC as the base with length, we already calculated as \(BC = 5\) knowing that base BC runs horizontally from B(4, 6) to C(7, 2).
- Height \(h\) = 6-2 = 4.
Now we can use the formula for the area of a triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times BC \times h = \frac{1}{2} \times 5 \times 4 = 10 \]
Final Responses
Now you can fill the areas with these values:
- Side Length AB: 5
- Side Length BC: 5
- Side Length AC: 6
- Perimeter of Triangle ABC: 16
- Area of Triangle ABC: 10
Make sure to replace the hypothetical coordinates and values with your triangle's actual coordinates and do the calculations as appropriate.
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