Question 1208635: Find the equation of a circle inscribed in a triangle with the sides on the lines x-3y=-5, 3x+y=1 and 3x-y=-11.
Found 3 solutions by Feliz_1965, ikleyn, math_tutor2020:
Answer by Feliz_1965(1)   (Show Source):
You can put this solution on YOUR website! 1. Find the Vertices of the Triangle.
The vertices of the triangle are the points where the lines intersect. We can find these by solving the system of equations in pairs:
Intersection of x - 3y = -5 and 3x + y = 1: Solving this system (e.g., using elimination or substitution), we get the point (-1, 2).
Intersection of x - 3y = -5 and 3x - y = -11: Solving this system, we get the point (-4, -1).
Intersection of 3x + y = 1 and 3x - y = -11: Solving this system, we get the point (-1, 4).
2. Find the Angle Bisectors.
The center of the inscribed circle (the incenter) is the point where the angle bisectors of the triangle intersect. To find an angle bisector, we can use the following:
A. Find the slopes of the two lines forming the angle.
B. Find the angle between the lines using the formula: tan(θ) = |(m1 - m2) / (1 + m_1*m_2)|, where m_1 and m_2 are the slopes.
The angle bisector will divide the angle θ in half.
Use the half-angle formula to find the slope of the bisector.
Use the point-slope form to find the equation of the bisector.
You only need to find two angle bisectors, as their intersection point will be the incenter.
3. Find the Incenter.
Solve the system of equations formed by the two angle bisectors you found. The solution is the incenter (h, k) of the circle.
4. Find the Radius.
The radius of the inscribed circle is the perpendicular distance from the incenter to any of the sides of the triangle.
Choose one of the side equations.
Use the formula for the distance from a point to a line: Distance = |Ax + By + C| / √(A² + B²) where (A, B, C) are the coefficients of the line equation, and (x, y) is the incenter.
5. Write the Equation of the Circle.
The standard form of the equation of a circle is: (x - h)² + (y - k)² = r²
Substitute the incenter (h, k) and the radius r that you found into this equation.
I hope this helps.
Answer by ikleyn(52782)  (Show Source):
You can put this solution on YOUR website! .
In the post by @Feliz_1965, this person writes
Intersection of x - 3y = -5 and 3x + y = 1: Solving this system (e.g., using elimination or substitution), we get the point (-1, 2).
Intersection of x - 3y = -5 and 3x - y = -11: Solving this system, we get the point (-4, -1).
Intersection of 3x + y = 1 and 3x - y = -11: Solving this system, we get the point (-1, 4).
No one of the given "solutions" is correct.
In other words, all three announced "solutions" are wrong.
You can easily check it by substituting the coordinates of these points,
that announced as the intersection points, into the corresponding equations.
Answer by math_tutor2020(3817)  (Show Source):
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