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Find the equation of a circle if it is inscribed in the triangle determined by the lines y = 0, 3x - 4y + 30 = 0, and 4x + 3y = 60.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| A circle that is inscribed in a triangle from the three lines. (Photo by Math Principles in Everyday Life) |
The first thing that we need to do is to get the equations of the angle bisectors of the given lines in order to get the coordinates of the center of a circle as well as its radius.
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| A circle that is inscribed in a triangle from the three lines. (Photo by Math Principles in Everyday Life) |
By using the distance of a point to the line formula, the angle bisector of lines y = 0 and 3x - 4y + 30 = 0 will be equal to
By using the distance of a point to the line formula, the angle bisector of lines y = 0 and 4x + 3y = 60 will be equal to
Hence, the angle bisectors are
Two angle bisectors are enough to use in order to get their intersection which is the center of a circle. Subtract the second equation from the first equation, we have
Substitute the value of y to either of the two equations above, we have
The coordinates of the center of a circle is C (5, 5). Since the center of a circle is located above the side of a triangle which is also the y-axis, then the radius of a circle is equal to 5.
Therefore, the equation of a circle is
