FIND THE MISSING SIDE LENGHTS,PERIMETER, AND AREA OF TRIANGLE ABC ON THE COORDINATE PLANE.ALL ANSWERS SHOULD BE IN SIMPLEST 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 ABC IS? AREA OF TRIANGLE ABC IS? ANSWER CHOICES ARE- SQRT 18, SQRT 13, SQRT 5, 5, 5+3SQRT2+SQRT13, 15SQRT26, 7.5 IF YOU CAN HELP THANK YOU.
To find the lengths of the sides of triangle ABC, we can use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
1. Length of side AB:
AB = sqrt((1 - 4)^2 + (2 - 5)^2)
= sqrt((-3)^2 + (-3)^2)
= sqrt(9 + 9)
= sqrt(18)
Therefore, the length of side AB is sqrt(18).
2. Length of side BC:
BC = sqrt((6 - 1)^2 + (2 - 2)^2)
= sqrt((5)^2 + (0)^2)
= sqrt(25 + 0)
= sqrt(25)
Therefore, the length of side BC is 5.
3. Length of side AC:
AC = sqrt((6 - 4)^2 + (2 - 5)^2)
= sqrt((2)^2 + (-3)^2)
= sqrt(4 + 9)
= sqrt(13)
Therefore, the length of side AC is sqrt(13).
4. Perimeter of triangle ABC:
Perimeter = AB + BC + AC
Perimeter = sqrt(18) + 5 + sqrt(13)
5. Area of triangle ABC:
To find the area of the triangle, we can use the Shoelace Formula or the determinant method.
For simplicity, let's use the Shoelace Formula:
Area = |(1/2)(x1y2 + x2y3 + x3y1 - x2y1 - x3y2 - x1y3)|
Area = |(1/2)((4*2) + (1*2) + (6*5) - (1*5) - (6*2) - (4*2))|
Area = |(1/2)(8 + 2 + 30 - 5 - 12 - 8)|
Area = |(1/2)(25)|
Area = 25/2
Therefore, the area of triangle ABC is 25/2.
All answers are in simplest radical form if necessary.