The definition of three noticeable points will be explored below.
To match the points with their definitions, the segments that form the points should be defined. Looking at the graph for point , it can be seen that the segments do not bisect the sides of the triangle. However, they bisect the interior angles of the triangle.
The segments bisect the sides of the triangle and connect the midpoints of the sides with their opposite vertices. Therefore, they are the medians of the triangle, and point is the centroid. Lastly, the segments that form point will be defined.
The concepts previously investigated can also be used in real life!
In the Mile High City, Denver, the cities transportation department is planning to pave three roads that connect three neighborhoods.
Magdalena and Vincenzo are the owners of two competing hotel chains. They see this as an opportunity to expand their empires into this region. Magdalena wants her hotel to be equidistant from each paved road. On the other hand, Vincenzo wants his hotel to be equidistant from the neighborhoods.
To follow another connection between circles and triangles, consider the circumcenter of a triangle. Recall that this point is equidistant from the vertices of the triangle. Therefore, a circle circumscribed at the triangle and centered at the circumcenter can be drawn.
At night, she wants to monitor all three gates. Therefore, she will place a lamp post in her farm. Where should she place the lamp so that each of the three corners are illuminated? Define the region illuminated by the light.
Consider the definitions of the circles of a triangle.
With the topics seen in this lesson, the challenge presented at the beginning can be answered. The circle that is tangent to each side of the triangle is the inscribed circle of the triangle. The circle that passes through the three vertices of the triangle is the circumscribed circle of the triangle.
For example, consider a carpenter designing a triangular table with one leg. To determine the location of the leg, he will use the centroid of the table. Since the centroid is the center of mass, the table will be perfectly balanced.