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.:
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
a is (1,4)
b is (-3,2)
c is (0,2)
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.
To solve this, let's first calculate the side lengths of triangle ABC using the distance formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) on a coordinate plane:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
**Side Length AB:**
We'll calculate the distance between points A (1,4) and B (-3,2) using the distance formula.
\[ AB = \sqrt{(-3 - 1)^2 + (2 - 4)^2} \]
\[ AB = \sqrt{(-4)^2 + (-2)^2} \]
\[ AB = \sqrt{16 + 4} \]
\[ AB = \sqrt{20} \]
\[ AB = \sqrt{4 \times 5} \]
\[ AB = 2\sqrt{5} \]
**Side Length BC:**
Now we'll find the distance between points B (-3,2) and C (0,2).
\[ BC = \sqrt{(0 - (-3))^2 + (2 - 2)^2} \]
\[ BC = \sqrt{3^2 + 0^2} \]
\[ BC = \sqrt{9} \]
\[ BC = 3 \]
**Side Length AC:**
Lastly, we'll calculate the distance between points A (1,4) and C (0,2).
\[ AC = \sqrt{(0 - 1)^2 + (2 - 4)^2} \]
\[ AC = \sqrt{(-1)^2 + (-2)^2} \]
\[ AC = \sqrt{1 + 4} \]
\[ AC = \sqrt{5} \]
**Perimeter of Triangle ABC:**
To find the perimeter of triangle ABC, we'll add all the side lengths together.
\[ \text{Perimeter} = AB + BC + AC \]
\[ \text{Perimeter} = 2\sqrt{5} + 3 + \sqrt{5} \]
\[ \text{Perimeter} = 3\sqrt{5} + 3 \]
**Area of Triangle ABC:**
For the area, we'll use the formula \( A = \frac{1}{2}bh \). Here, BC can serve as the base, and the height (h) can be the difference in the y-coordinates of points A and C since A and C are vertically aligned.
The height h is the difference in the y-coordinates of A and C:
\[ h = |y_A - y_C| \]
\[ h = |4 - 2| \]
\[ h = 2 \]
The base is the length of side BC which we've already calculated as 3.
Now use the area formula \( A = \frac{1}{2}bh \):
\[ A = \frac{1}{2} \times 3 \times 2 \]
\[ A = \frac{1}{2} \times 6 \]
\[ A = 3 \]
Therefore, the side lengths, perimeter, and area of triangle ABC are:
- Side Length AB: \( 2\sqrt{5} \)
- Side Length BC: \( 3 \)
- Side Length AC: \( \sqrt{5} \)
- Perimeter of Triangle ABC: \( 3\sqrt{5} + 3 \)
- Area of Triangle ABC: \( 3 \)