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:

point A is at (-1,4)

point B is at (-3,2)

point C is at (0,2)


Side Length AB: Response area

Side Length BC: Response area

Side Length AC: Response area

Perimeter of Triangle ABC: Response area

Area of Triangle ABC:

Answer 1

To find the missing side lengths, perimeter, and area of triangle ABC, we will first calculate the lengths of each side using the distance formula and then compute the perimeter and area.

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Calculate side lengths

**1. Side Length AB:**
- Points A and B are at \((-1, 4)\) and \((-3, 2)\).

\[
AB = \sqrt{((-3) - (-1))^2 + ((2) - (4))^2}
= \sqrt{(-2)^2 + (-2)^2}
= \sqrt{4 + 4}
= \sqrt{8}
= 2\sqrt{2}
\]

**2. Side Length BC:**
- Points B and C are at \((-3, 2)\) and \((0, 2)\).

\[
BC = \sqrt{(0 - (-3))^2 + (2 - 2)^2}
= \sqrt{(3)^2 + (0)^2}
= \sqrt{9}
= 3
\]

**3. Side Length AC:**
- Points A and C are at \((-1, 4)\) and \((0, 2)\).

\[
AC = \sqrt{(0 - (-1))^2 + (2 - 4)^2}
= \sqrt{(1)^2 + (-2)^2}
= \sqrt{1 + 4}
= \sqrt{5}
\]

### Step 2: Calculate perimeter

The perimeter \(P\) of triangle ABC is the sum of the lengths of its sides:

\[
P = AB + BC + AC
= 2\sqrt{2} + 3 + \sqrt{5}
\]

### Step 3: Calculate area

To find the area of triangle ABC, we can use the formula for the area of a triangle given vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

\[
\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|
\]

Substituting the points \(A(-1, 4)\), \(B(-3, 2)\), and \(C(0, 2)\):

\[
\text{Area} = \frac{1}{2} |-1(2 - 2) + (-3)(2 - 4) + 0(4 - 2)|
\]
\[
= \frac{1}{2} |0 + (-3)(-2) + 0|
\]
\[
= \frac{1}{2} |6| = 3
\]

### Summary of results

- **Side Length AB:** \(2\sqrt{2}\)
- **Side Length BC:** \(3\)
- **Side Length AC:** \(\sqrt{5}\)
- **Perimeter of Triangle ABC:** \(2\sqrt{2} + 3 + \sqrt{5}\)
- **Area of Triangle ABC:** \(3\)

These results can now be documented explicitly:

**Side Length AB:** \(2\sqrt{2}\)

**Side Length BC:** \(3\)

**Side Length AC:** \(\sqrt{5}\)

**Perimeter of Triangle ABC:** \(2\sqrt{2} + 3 + \sqrt{5}\)

**Area of Triangle ABC:** \(3\)