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 (-1,4)
point b (-3,2)
point c (0,2)

(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.

Answer 1

To find the side lengths, perimeter, and area of triangle ABC with the given points A(-1, 4), B(-3, 2), and C(0, 2), we first calculate the side lengths using the distance formula.

### 1. Calculate the lengths of the sides of the triangle

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

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

#### Side Length AB:
Points: A(-1, 4) and B(-3, 2)

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

#### Side Length BC:
Points: B(-3, 2) and C(0, 2)

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

#### Side Length AC:
Points: A(-1, 4) and C(0, 2)

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

### 2. Calculate the perimeter of Triangle ABC
The perimeter \(P\) is the sum of all side lengths:

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

### 3. Calculate the area of Triangle ABC
The area \(A\) can be calculated using the formula:

\[
A = \frac{1}{2} \times b \times h
\]

Where \(b\) is the length of base \(BC\) and \(h\) is the height from point \(A\) to line segment \(BC\). Here, base \(BC = 3\).

To find the height, we can observe that points B and C share the same y-coordinate (y=2). The height from point A(-1, 4) to base BC is the vertical distance from A to y=2:

\[
h = 4 - 2 = 2
\]

Now substituting the base \(b = 3\) and height \(h = 2\) into the area formula:

\[
A = \frac{1}{2} \times 3 \times 2 = 3
\]

### 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: \(3\)

Now, filling the responses in the requested format:

**Responses:**
- 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: \(3\)