FIND THE MISSING SIDE LENGHTS, PERIMETER, AND AREA OF TRIANGLE ABC ON THE COORDINATE PLANE. ALL ANSWERS 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 ABC IS? AREA OF TRIANGLE ABC IS? POSSIBLE ANSWERS ARE SQRT 18, SQRT 13, SQRT 5, 5, 5+3SQRT2+SQRT13, 15SQRT26 AND 7.5. YOU GAVE AN ANSWER EARLIER OF 25/2 FOR THE AREA OF TRIANGLE BUT THAT ANSWER IS NOT ON THE LIST OF POSSIBLE ANSWERS, AND ONE POSSIBLE ANSWER I HAVE FOR THE AREA OF PRIMETER THAT IS CLOSE TO THE ANSWER YOU GAVE IS 5+3SQRT2+SQRT13 BUT IT,S NOT THE ANSWER YOU GAVE. THANK YOU.
To find the lengths of the sides of triangle ABC, we can use the distance formula:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
BC = sqrt((x3 - x2)^2 + (y3 - y2)^2)
AC = sqrt((x3 - x1)^2 + (y3 - y1)^2)
Using the given coordinates:
A=(4,5)
B=(1,2)
C=(6,2)
AB = sqrt((1 - 4)^2 + (2 - 5)^2) = sqrt((-3)^2 + (-3)^2) = sqrt(18)
BC = sqrt((6 - 1)^2 + (2 - 2)^2) = sqrt((5)^2 + (0)^2) = sqrt(25) = 5
AC = sqrt((6 - 4)^2 + (2 - 5)^2) = sqrt((2)^2 + (-3)^2) = sqrt(4 + 9) = sqrt(13)
Therefore, side length AB is sqrt(18), side length BC is 5, and side length AC is sqrt(13).
To find the perimeter of triangle ABC, we can add the lengths of all three sides:
Perimeter = AB + BC + AC = sqrt(18) + 5 + sqrt(13)
To find the area of triangle ABC, we can use the formula for the area of a triangle:
Area = 1/2 * base * height
We can take AB as the base and the perpendicular distance from point C to line AB as the height. To find the perpendicular distance, we can use the formula:
Height = |(x3 - x1)(y2 - y1) - (x2 - x1)(y3 - y1)| / 2
Plugging in the values:
Height = |(6 - 4)(2 - 5) - (1 - 4)(2 - 5)| / 2
= |-2(3) + (-3)(-3)| / 2
= |-6 + 9| / 2
= 3 / 2 = 1.5
Area = 1/2 * AB * Height
= 1/2 * sqrt(18) * 1.5
= 3/4 * sqrt(18)
= 3/4 * sqrt(9 * 2)
= 3/4 * 3 * sqrt(2)
= 9/4 * sqrt(2)
= (9/4)sqrt(2)
Therefore, the perimeter of triangle ABC is sqrt(18) + 5 + sqrt(13) and the area of triangle ABC is (9/4)sqrt(2).