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:

Question

A (-1, 4)
B (-3,2)
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 missing side lengths, perimeter, and area of triangle ABC given the points A (-1, 4), B (-3, 2), and C (0, 2), we'll follow these steps:

Step 1: Calculate the side lengths

Side Length AB: Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \):

• 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:

• 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:

• A(-1, 4) and C(0, 2): \[ AC = \sqrt{(0 - (-1))^2 + (2 - 4)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

Step 2: Calculate the perimeter of triangle ABC

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

Step 3: Calculate the area of triangle ABC

Using the formula \( A = \frac{1}{2} b h \):

• Base (BC) = 3
• Height (h) is the y-distance from point A to line BC (which is y = 2). The height is the vertical distance from A(4) to line BC(2):
  \[ h = 4 - 2 = 2 \] \[ \text{Area} = \frac{1}{2} \times 3 \times 2 = 3 \]

Final 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: \( 3 \)

You can now insert these responses accordingly.