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 = (4,5)
b is: B = (1,2)
c is: C = (6,2)
Note: the formula for 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:
√18
9√2
3√2
√4 + √9
√13
√5
5
5 + √13
5 + 3√2 + √13
15√26
7.5
15
To find the side lengths of triangle ABC, we can use the distance formula. The distance formula is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Side length AB:
Using points A(4,5) and B(1,2), we can plug in the values into the distance formula:
dAB = √((1 - 4)^2 + (2 - 5)^2)
= √((-3)^2 + (-3)^2)
= √(9 + 9)
= √18
So, the length of AB is √18.
Side length BC:
Using points B(1,2) and C(6,2), we can plug in the values into the distance formula:
dBC = √((6 - 1)^2 + (2 - 2)^2)
= √((5)^2 + (0)^2)
= √(25 + 0)
= √25
= 5
So, the length of BC is 5.
Side length AC:
Using points A(4,5) and C(6,2), we can plug in the values into the distance formula:
dAC = √((6 - 4)^2 + (2 - 5)^2)
= √((2)^2 + (-3)^2)
= √(4 + 9)
= √13
So, the length of AC is √13.
Perimeter of triangle ABC:
The perimeter is the sum of all the side lengths:
Perimeter = AB + BC + AC
= √18 + 5 + √13
Area of triangle ABC:
To find the area, we can use the formula A = bh/2. The base (BC) is 5, and we need to find the height, which is the distance from point A to side BC.
Using the distance formula with points A(4,5) and the line containing side BC (which has endpoints (1,2) and (6,2)), we can write the equation of the line:
y = 2
To find the height, we need to find the perpendicular distance from point A to the line y = 2. This distance is the difference between the y-coordinate of point A and the y-coordinate of the line y = 2:
h = 5 - 2
= 3
Now we can use the formula A = bh/2:
Area = 5 * 3/2
= 15/2
So, the area of triangle ABC is 15/2 or 7.5.
In summary:
Side length AB: √18
Side length BC: 5
Side length AC: √13
Perimeter of triangle ABC: √18 + 5 + √13
Area of triangle ABC: 15/2 or 7.5