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: (5 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. 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: Response area Note: the formula for area of a triangle is A=bh2 , where b is the length of the base (BC) and h is the height from side BC to point A. A = -1, 4 B= -3, 2 C= 0,2

Answer 1

To find the missing side lengths, perimeter, and area of triangle ABC with vertices A(-1, 4), B(-3, 2), and C(0, 2), we can calculate the lengths of the sides using the distance formula and then find the perimeter and area.

### 1. Calculate Side Length AB
Using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\):
\[
AB = \sqrt{((-3) - (-1))^2 + ((2) - (4))^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

### 2. Calculate Side Length BC
\[
BC = \sqrt{(0 - (-3))^2 + ((2) - (2))^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3
\]

### 3. Calculate Side Length AC
\[
AC = \sqrt{(0 - (-1))^2 + ((2) - (4))^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

### 4. Calculate Perimeter
The perimeter \(P\) of triangle ABC is the sum of the side lengths:
\[
P = AB + BC + AC = 2\sqrt{2} + 3 + \sqrt{5}
\]

### 5. Calculate Area
To find the area using the formula \(A = \frac{1}{2}bh\), we need to determine the base (BC) and height (h from A to BC):

1. **Base BC**: We already computed \(BC = 3\).
2. **Height**: The line segment BC is horizontal (y = 2) because B and C have the same y-coordinate of 2. The height from point A (-1, 4) to line BC (y = 2) is the vertical distance, which is:
\[
h = 4 - 2 = 2
\]

Now, plug these values into the area formula:
\[
A = \frac{1}{2} \cdot BC \cdot h = \frac{1}{2} \cdot 3 \cdot 2 = 3
\]

Putting all results together:

- **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**: \(3\)

Now, placing the calculated values in the correct input areas would look like this:

**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**: \(3\)