FIND THE MISSING SIDE LENGHTS, PERIMETER, AND AREA OF TRIANGLE ABC ON THE COORDINATE PLANE. ALL AMSWERS SHOULD BE IN SIMPLIST RADICAL FORM IF NECESSARY. A= (4,5) B=(1,2) C=(6,2). SIDE LENGHT AB IS? SIDE LENGHT BC IS? SIDE LENGHT AC IS? PERIMETER OF TRIANGLE IS? AREA OF TRIANGLE IS?
To find the lengths of the sides, we can use the distance formula:
The distance between two points (x1,y1) and (x2,y2) is given by the formula:
d = √((x2-x1)^2 + (y2-y1)^2)
1. Length of AB:
AB = √((1-4)^2 + (2-5)^2)
= √((-3)^2 + (-3)^2)
= √(9 + 9)
= √18
= 3√2
2. Length of BC:
BC = √((6-1)^2 + (2-2)^2)
= √(5^2 + 0^2)
= √25
= 5
3. Length of AC:
AC = √((6-4)^2 + (2-5)^2)
= √(2^2 + (-3)^2)
= √(4 + 9)
= √13
The perimeter of the triangle ABC is the sum of the lengths of its sides:
Perimeter = AB + BC + AC
= 3√2 + 5 + √13
To find the area of the triangle, we can use the formula:
Area = 1/2 * base * height
The base and height can be found by using the coordinates of the triangle.
The base of the triangle is the length of BC, which is 5.
The height can be found by calculating the perpendicular distance from point A to line BC. To do this, we can find the equation of the line passing through points B and C, and then find the distance from point A to that line.
Let's find the equation of the line BC:
Slope of BC = (2-2)/(6-1) = 0/5 = 0
Since the slope is 0, the line BC is a horizontal line passing through y = 2.
The height is the perpendicular distance from point A to the line y = 2. It can be calculated as the absolute difference between the y-coordinate of point A and the y-coordinate of point B or C, which is 5 - 2 = 3.
Area = 1/2 * base * height
= 1/2 * 5 * 3
= 15/2
= 7.5 square units (approximate)
So, the side lengths, perimeter, and area of triangle ABC are:
AB = 3√2
BC = 5
AC = √13
Perimeter = 3√2 + 5 + √13
Area = 7.5 square units (approximate).