Find the equation of a circle circumscribes a triangle determined by the line y= 0 , y= x and 2x+3y= 10
I beg you
To find the equation of a circle that circumscribes a triangle determined by three lines, we need to find the coordinates of the triangle's three vertices.
Given the lines:
1. y = 0
2. y = x
3. 2x + 3y = 10
First, let's find the intersection point of lines 1 and 2.
By substituting y = x into y = 0, we get:
x = 0
So, the first vertex is (0, 0).
Next, let's find the intersection of lines 1 and 3.
Substituting y = 0 into 2x + 3y = 10, we have:
2x = 10
x = 5
So, the second vertex is (5, 0).
Finally, let's find the intersection of lines 2 and 3.
Substituting y = x into 2x + 3y = 10, we have:
2x + 3x = 10
5x = 10
x = 2
Substituting x = 2 into y = x, we have:
y = 2
So, the third vertex is (2, 2).
Now that we have the coordinates of the triangle's three vertices: (0, 0), (5, 0), and (2, 2), we can find the center and radius of the circumscribing circle.
The center of the circle is the intersection of the perpendicular bisectors of any two sides of the triangle.
Let's consider the sides determined by the vertices (0, 0) and (5, 0) and by the vertices (2, 2) and (0, 0).
The midpoint of the first side is ((0+5)/2, (0+0)/2) = (2.5, 0).
Its slope is (0 - 0) / (5 - 0) = 0.
The slope of a line perpendicular to this side would be negative reciprocal of 0, which is undefined. However, if we extend the line y = 0 infinitely, we can consider it as a vertical line.
So, the perpendicular bisector of the side determined by (0, 0) and (5, 0) is the vertical line x = 2.5.
The midpoint of the second side is ((2+0)/2, (2+0)/2) = (1, 1).
The slope of the second side is (0 - 2) / (5 - 0) = -2/5.
The slope of the line perpendicular to this side would be the negative reciprocal, which is 5/2.
Using the point-slope form, we have the equation of the perpendicular bisector:
y - 1 = (5/2)(x - 1)
2y - 2 = 5x - 5
5x - 2y = 3
Now, we need to find the intersection of the lines x = 2.5 and 5x - 2y = 3.
Substituting x = 2.5 into 5x - 2y = 3, we get:
12.5 - 2y = 3
-2y = 3 - 12.5
-2y = -9.5
y = -9.5 / -2
y = 4.75
So, the center of the circumscribing circle is (2.5, 4.75).
To find the radius, we can calculate the distance between one of the triangle vertices and the center of the circle.
Let's consider the distance between (0, 0) and (2.5, 4.75).
Using the distance formula:
r = √[(x2 - x1)^2 + (y2 - y1)^2]
r = √[(2.5 - 0)^2 + (4.75 - 0)^2]
r = √[6.25 + 22.5625]
r = √28.8125
r = 5.36 (approximately)
Therefore, the equation of the circle that circumscribes the given triangle is:
(x - 2.5)^2 + (y - 4.75)^2 = 28.8125