if you have the triangle ABC with the points A(47,100) B(-1273,203) and C(971,-732) how would you find the center and the radius of the incircle of triangle ABC?
To find the center and radius of the incircle of triangle ABC, we need to follow these steps:
Step 1: Find the lengths of the triangle's sides.
To find the lengths of the triangle's sides, we can use the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
In this case, the sides of the triangle are AB, BC, and CA. Using the given coordinates of points A, B, and C, we can calculate the lengths of these sides.
AB:
Distance AB = √[(-1273 - 47)^2 + (203 - 100)^2]
BC:
Distance BC = √[(971 - (-1273))^2 + (-732 - 203)^2]
CA:
Distance CA = √[(971 - 47)^2 + (-732 - 100)^2]
Step 2: Calculate the semi-perimeter of the triangle.
The semi-perimeter (s) is the sum of the lengths of the triangle's sides divided by 2.
s = (AB + BC + CA) / 2
Step 3: Find the area of the triangle.
The area (A) of the triangle can be calculated using Heron's formula:
A = √[s * (s - AB) * (s - BC) * (s - CA)]
Step 4: Calculate the radius (r) of the incircle.
The radius of the incircle can be found using the formula:
r = A / s
Step 5: Find the coordinates of the incenter.
The incenter of a triangle is the intersection point of the triangle's angle bisectors. To find the coordinates (x, y) of the incenter, we can use the formula:
x = (xA * A + xB * B + xC * C) / (A + B + C)
y = (yA * A + yB * B + yC * C) / (A + B + C)
Where (xA, yA) are the coordinates of point A, (xB, yB) are the coordinates of point B, (xC, yC) are the coordinates of point C, and A, B, C are the lengths of the opposite sides.
Once we have the center coordinates (x, y) and the radius (r), we can determine the equation of the incircle using the equation: (x - xC)^2 + (y - yC)^2 = r^2.