Circle Inscribed - Triangle Problems

Category: Plane Geometry

"Published in Newark, California, USA"

The base of an isosceles triangle is 16 in. and the altitude is 15 in. Find the radius of the inscribed circle.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The area of ∆ABC is

Photo by Math Principles in Everyday Life

By using Pythagorean Theorem, we can solve for the two legs of an isosceles triangle as follows

Next, draw the angle bisectors of an isosceles traingle as follows

Photo by Math Principles in Everyday Life

The intersection of the angle bisectors of an isosceles triangle is the center of an inscribed circle which is point O. From point O, draw a line which is perpendicular to AB, draw a line which is perpendicular to AC, and draw a line which is perpendicular to BC. These three lines will be the radius of a circle. 

Photo by Math Principles in Everyday Life

The total area of an isosceles triangle is equal to the area of three triangles whose vertex is point O. Therefore, the radius of an inscribed circle is

Finding Equation - Circle, 11

Category: Analytic Geometry, Plane Geometry, Algebra

"Published in Vacaville, California, USA"

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

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. 

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