Question
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
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
Answers
GPT 3.5
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)
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)