Question

Find the missing side lengths, perimeter, and area of triangle ABC on the coordinate plane. All answers should be in simplest radical form, if necessary: Side Length AB: Side Length BC: Side Length AC: Perimeter of Triangle ABC: Area of Triangle ABC: Note the formula for the area of a triangle , where b is the length of the base (BC) and h is the height BC to point A. A.(1,4) B. (0,2) C.(3,2)

Answer 1

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.