To find the lengths of the sides of triangle ABC and subsequently the perimeter and area, we'll use the distance formula and the properties of triangles.
Step 1: Find the lengths of the sides using the distance formula
Given the coordinates of the points:
- A(1, 4)
- B(0, 2)
- C(3, 2)
The distance formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
1. Length of AB:
\[ AB = \sqrt{(0 - 1)^2 + (2 - 4)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
2. Length of BC:
\[ BC = \sqrt{(3 - 0)^2 + (2 - 2)^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3 \]
3. Length of AC:
\[ AC = \sqrt{(3 - 1)^2 + (2 - 4)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
Step 2: Calculate the perimeter of triangle ABC
The perimeter \( P \) is the sum of all the sides: \[ P = AB + BC + AC = \sqrt{5} + 3 + 2\sqrt{2} \]
Step 3: Calculate the area of triangle ABC
To find the area, we can use the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Here the base \( BC \) is 3. The height is the vertical distance from A to line BC.
The line segment \( BC \) is horizontal at \( y = 2 \) (since both B and C have \( y \)-coordinate of 2).
The height from point A(1, 4) to line \( BC \) (at \( y=2 \)) is: \[ \text{height} = 4 - 2 = 2 \]
Now compute the area: \[ \text{Area} = \frac{1}{2} \times 3 \times 2 = \frac{3 \times 2}{2} = 3 \]
Summary of Results
- Side Length AB: \(\sqrt{5}\)
- Side Length BC: \(3\)
- Side Length AC: \(2\sqrt{2}\)
- Perimeter of Triangle ABC: \(\sqrt{5} + 3 + 2\sqrt{2}\)
- Area of Triangle ABC: \(3\)
All results are in simplest radical form.
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