Give the CORRECT ANSWERS and 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 side lengths of the triangle, we can use the distance formula. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Side length AB:
Using the coordinates (-1, 4) for point A and (-3, 2) for point B:
d = √((-3 - (-1))^2 + (2 - 4)^2)
d = √((-3 + 1)^2 + (2 - 4)^2)
d = √((-2)^2 + (-2)^2)
d = √(4 + 4)
d = √8
d = 2√2
Side length BC:
Using the coordinates (-3, 2) for point B and (0, 2) for point C:
d = √((0 - (-3))^2 + (2 - 2)^2)
d = √((0 + 3)^2 + 0^2)
d = √(3^2 + 0^2)
d = √(9 + 0)
d = √9
d = 3
Side length AC:
Using the coordinates (-1, 4) for point A and (0, 2) for point C:
d = √((0 - (-1))^2 + (2 - 4)^2)
d = √((0 + 1)^2 + (-2)^2)
d = √(1^2 + 4)
d = √(1 + 4)
d = √5
Perimeter of triangle ABC:
Perimeter = AB + BC + AC
Perimeter = 2√2 + 3 + √5
To find the area of triangle ABC, we can use the formula A = bh / 2, where b is the length of the base BC and h is the height from side BC to point A.
Base BC = 3
To find the height, we need to find the length of the perpendicular line from point A to side BC. The slope of the line BC is 0 since the y-coordinates of B and C are the same. The slope of the perpendicular line to BC passing through point A is the negative reciprocal of the slope of BC, which is undefined.
This means the height is the vertical distance from point A to line BC, which is the difference between the y-coordinate of point A and the y-coordinate of point B or C.
Height = 4 - 2 = 2
Area of triangle ABC:
A = (base)(height) / 2
A = (3)(2) / 2
A = 6 / 2
A = 3
Side length AB: 2√2
Side length BC: 3
Side length AC: √5
Perimeter of triangle ABC: 2√2 + 3 + √5
Area of triangle ABC: 3
a is: A = (-1,4)
b is: B = (-3,2)
c is: C = (0,2)