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.

Answer 1

To solve for the side lengths, perimeter, and area of triangle ABC on the coordinate plane, we need the coordinates of the vertices A, B, and C. Assuming we can define these coordinates, let's denote the points A(x1, y1), B(x2, y2), and C(x3, y3).

Let's go through the calculations step by step.

### Step 1: Calculate Side Lengths
The length of each side can be calculated using the distance formula:

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

1. **Side AB**
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

2. **Side BC**
\[
BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}
\]

3. **Side AC**
\[
AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2}
\]

### Step 2: Calculate the Perimeter
The perimeter \( P \) of triangle ABC is given by:
\[
P = AB + BC + AC
\]

### Step 3: Calculate the Area
If we take BC as the base and the height as the perpendicular distance from point A to the line BC, we can use the area formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

If the coordinates of points and heights are not provided, we’d compute the height using coordinates or any additional information about the triangle.

### Assume example coordinates as below for demonstration:

Let:
- A(1, 2)
- B(4, 6)
- C(5, 3)

Using the coordinates, we calculate:

1. **Side Lengths**:
- \( AB = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
- \( BC = \sqrt{(5 - 4)^2 + (3 - 6)^2} = \sqrt{1 + 9} = \sqrt{10} \)
- \( AC = \sqrt{(5 - 1)^2 + (3 - 2)^2} = \sqrt{16 + 1} = \sqrt{17} \)

2. **Perimeter**:
\[
P = 5 + \sqrt{10} + \sqrt{17}
\]

3. **Area**: To find the area, we may need to find the perpendicular height from A to BC. For simplicity, let's assume the perpendicular height \( h \) is determined to be 3.
\[
\text{Area} = \frac{1}{2} \times BC \times h
\]
Where \( BC \) is the base, giving this expression:
Using a real calculation needs values of h.

Finally, you input the correct values based on coordinate calculations in the respective boxes:

### Input Responses:

- Side Length AB: \( 5 \)
- Side Length BC: \( \sqrt{10} \)
- Side Length AC: \( \sqrt{17} \)
- Perimeter of Triangle ABC: \( 5 + \sqrt{10} + \sqrt{17} \)
- Area of Triangle ABC: (needs actual h to give exact numeric output)

Please replace the assumed coordinates with the actual triangle vertices you have to complete your problem correctly.