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Explore

Investigating Medians of a Triangle

Using the following applet, draw the medians of

△ A B C .

Drawing medians of triangle ABC

Do the medians of the triangle intersect in only one point? If yes, calculate the ratio

\frac{a}{b},

where

a

is the distance from a vertex to the point of intersection of the medians and

b

is the total length of the median drawn from the vertex. What conjectures can be made based on those results?

Discussion

Centroid and Centroid Theorem

As it can be seen, medians of a triangle meet at a point. This point has a special name.

By means of exploring the applet, it was also seen that the ratio $\frac{a}{b}$ is constant for each median.

Example

Triangular Table

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.

Triangular table with a 6 feet long median

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.

Answer

See solution.

Hint

Begin by finding the midpoint of one of the sides using a ruler. Then draw the median of the side to determine the centroid of the table.

Solution

Recall that the centroid of a triangle is also the center of mass of the triangle. Therefore, if Zosia places the leg at the centroid, the table will be perfectly balanced. Using a ruler, Zosia can begin finding the centroid by drawing the median of a table side.

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?

\frac{2}{3} \cdot 6 = 4

The centroid is $4$ feet away from the vertex.

Therefore, if Zosia places the leg at this point, the table will be perfectly balanced.

Explore

Investigating Perpendicular Bisectors of a Triangle

Use the applet to draw the perpendicular bisectors of the sides of

△ A B C .

Drawing the perpendicular bisectors of a triangle

Do the perpendicular bisectors of the sides of the triangle intersect at the same point? If yes, what can be concluded about the distances from each vertex to the point of intersection of the perpendicular bisectors?

Discussion

Circumcenter and Circumcenter Theorem

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.

Explore

Investigating Angle Bisectors of a Triangle

This time, draw the angle bisectors of the interior angles of

△ A B C .

Drawing angle bisectors of triangle ABC

Examine the location where the angle bisectors intersect. Is there only one point of intersection? If so, find out how far each side of

△ A B C

is from the point of intersection.

Discussion

Incenter and Incenter Theorem

Through exploration of the applet, it has been shown that the angle bisectors of a triangle intersect at one point.

Example

Setting a Table

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?

Hint

Note that the candle should be equidistant from each corner of the table.

Solution

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.

Explore

Investigating Altitudes of a Triangle

After understanding the characteristics of the bisectors of a triangle, the relationship between the altitudes of a triangle will be investigated. Using the applet, explore what properties are related to the altitude. Begin by drawing the altitudes of

△ A B C .

Drawing altitudes of triangle ABC

What can be assumed about the altitudes of a triangle?

Discussion

Orthocenter and the Orthocenter Theorem

As could be found in the previous exploration, when the altitudes of a triangle are drawn, they intersect at one point.

Closure

Euler's Line

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?

In any triangle, the centroid, circumcenter, and orthocenter of the triangle are collinear.

Showing collinearity of centroid, circumcenter, and orhtocenter of triangle ABC

The line passes through these points is called the Euler's line.

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