A L.

asked • 08/22/12# how to find the distance of the incenter of an equlateral triangle to mid center of each side?

I am given a diagram of a equalteral triangle with all bisectors drawn, the incenter and the circumcenter are one and the same. I am also given the distance of the circum center from the vertexes. I am told to find the distance from the incenter to the midpoint of each side, how do I solve it?

## 5 Answers By Expert Tutors

Susan H. answered • 08/23/12

High School/College level Math Tutor

Lisa M. answered • 08/23/12

3rd-7th English and math. Discount for IEPs.

The distance of the circumcenter to the vertex is two thirds of the altitude. The length you are trying to find is the other third. Just halve the distance you already know.

An equilateral triangle has 3 equal sides and 3 equal angles. Since the sum total of any triangle is 180 degrees, then each angle is 1/3 x 180 = 60 degrees. With the triangle having one side as the bottom or the base, drop a line vertically down from the triangle center (point C) to the bottom at the point which bisects that bottom side (point B). Draw another line from the center (C) to the angle at the left, (angle or point A). ABC is now a triangle with angle A = 30 degrees, angle B = 90 degrees, and Angle C as 60 degrees. The sides of this smaller triangle ABC are in a ratio of AC/CB = 2/1 AB/CB = 3^{1/2}/1 If the hypotenuse = 2, then CB = 1 Let a side of the equilateral triangle = x. Then AB=x/2. The distance from the center of the equilateral triangle to the bisection of a side = CB. AB/CB = (x/2)/CB = 3^{1/2}/1 = 3^{1/2 }Solve for CB = (x/2)/3^{1/2 }= x/2(3)^{1/2}. CB = half of the equilateral triangle's side divided by the square root of 3. For example, if a side of the equilateral triangle is 2, then the distance from its center to the bisection of the bottom is 1/2 times 2 = 1 divided by the square root of 3. CB would then be 1/3^{1/2}.

Adam B. answered • 08/23/12

Engineering Graduate specializing in Mathematics and Science

If the triangle is equilateral, then you form a 30, 60, 90 triangle. Use the bisector to the vertex as a hypotenuse with length equal to two times the length of the shortest side. The shortest sideis the distance from the circumcenter to the the vertexes. All you have to do is take the distance given to you and divide it by two.

Andres M. answered • 08/22/12

College/High School Math, Physical Sciences, and Spanish.

The first thing you have to keep in mind is the symmetry of the problem. Here you have an equiliateral triangle, meaning all sides have the same length and all angles measure the same. That gives you the first clue, which is that the angles of the triangle are all 60 degrees (360/3).

The second thing you have to realize is that any line from a vertex to the circumcenter will bisect the angle of 60 degrees into two 30 degree angles. Put this together with the fact that a line that goes through the midpoint of one of the sides of the triangle and the circumcenter.

If you draw all this out you will find out that you wind up with a nice set up to the problem, where you are looking for the length of one of the sides of a right triangle.

The final step is to use trigonometry. You'll have a right triangle with an angle of 30 degrees, and a hypotenuse equal in length to the distance between the circumcenter and the vertex. Solve for the side they are asking about (think about the sine and cosine definitions), and you should be set.

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Andres M.

Oops, i meant 180/3 = 60.

08/22/12