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:
Side length BC:
Side length AC:
Perimeter of triangle ABC:
Area of triangle ABC:
a is: A = (-1,4)
b is: B = (-3,2)
c is: C = (0,2)
Note: the formula for the area of a triangle is A = bh/2, where b is the length of the base (BC) and h is the height from side BC to point A.
Answer choices
A. 2√2
B. 2
C. 3
D. √3
E. √5
F. 5
G. √8
H. 3 + 2√2 + √5
I. 7 + √8
To find the missing side lengths, we can use the distance formula between two points in the coordinate plane:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can find the lengths of each side:
AB:
x1 = -1, y1 = 4
x2 = -3, y2 = 2
AB = √((-3 - (-1))^2 + (2 - 4)^2)
AB = √(4 + 4)
AB = √8 = 2√2
BC:
x1 = -3, y1 = 2
x2 = 0, y2 = 2
BC = √((0 - (-3))^2 + (2 - 2)^2)
BC = √(9 + 0)
BC = √9 = 3
AC:
x1 = -1, y1 = 4
x2 = 0, y2 = 2
AC = √((0 - (-1))^2 + (2 - 4)^2)
AC = √(1 + 4)
AC = √5
To find the perimeter, we add up the lengths of all three sides:
Perimeter of triangle ABC = AB + BC + AC
Perimeter of triangle ABC = 2√2 + 3 + √5
To find the area, we can use the formula A = bh/2. We need to find the height from side BC to point A.
The slope of BC can be determined by finding the difference in y-coordinates and the difference in x-coordinates:
m = Δy/Δx
m = (2 - 2)/(0 - (-3))
m = 0/3 = 0
Since the slope of BC is 0, the line is horizontal. Therefore, the height from side BC to point A is the difference in y-coordinates:
Height = 4 - 2 = 2
Now we can find the area:
Area of triangle ABC = (BC * Height)/2
Area of triangle ABC = (3 * 2)/2
Area of triangle ABC = 3
Therefore, the missing side lengths, perimeter, and area of triangle ABC are:
AB = 2√2
BC = 3
AC = √5
Perimeter of triangle ABC = 2√2 + 3 + √5
Area of triangle ABC = 3
The correct answer is H. 3 + 2√2 + √5