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Zosia is planning to throw a party in her new house. She wants to design a triangular table with one leg for the snacks and drinks. This design choice will ensure that no one while moving around, would bump into a table leg.
But wait, there is a problem she has to solve. She has no idea where to place the leg so that the table will be perfectly balanced. Lend some math skills and help her find the point on the table where the table leg should be placed.
See solution.
Since the centroid of a triangle is the point of intersection of the medians, the centroid will be on this median.
The Centroid Theorem states that the centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side. Using this theorem, the distance between the centroid and the vertex along the segment can be found. Notice that the length of the median is six feet. What is two-thirds of six feet?
The centroid is $4$ feet away from the vertex.
Therefore, if Zosia places the leg at this point, the table will be perfectly balanced.
The perpendicular bisectors of the sides of a triangle are concurrent, as shown in the preceding exploration. The point of concurrency of the perpendicular bisectors of a triangle is known by a unique name.
The investigation also indicated that the distances from the circumcenter to each vertex of the triangle are equal.
Through exploration of the applet, it has been shown that the angle bisectors of a triangle intersect at one point.
Now that Zosia has perfectly balanced her triangular table using the centroid, she is ready to put some snacks on it. The snacks should be equidistant from each side of the table so that her friends can reach them easily. To top it all off, Zosia wants to place a candle to illuminate the whole table. ¡Qué genial!
Where should she place the candle and snacks?
Note that the candle should be equidistant from each corner of the table.
Since the snacks should be equidistant from each side of the table, begin by recalling the Incenter Theorem.
Incenter Theorem |
The incenter of a triangle is the point which is equidistant from each of the triangle's sides. This point is considered to be the center of the triangle. |
By this theorem, it can be concluded that the snacks should be placed in the incenter of the table. On the other hand, the candle should be equidistant from each corner of the table to illuminate the whole table. Therefore, consider the Circumcenter Theorem.
Circumcenter Theorem |
The circumcenter of a triangle is the point which is equidistant from each of the triangle's vertices. |
Therefore, the candle illuminates the whole table if Zosia place it in the circumcenter of the table. Note that the centroid of the table does not satisfy either of these locations because it helps to determine the location of the center of mass as in the previous example.
In this lesson, relationships within triangles were covered. Four points have been introduced, along with their theorems. Did you know that there is a mysterious relationship between three of these points?