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Congruence, Proof, and Constructions

More Theorems About Triangles

In the previous lesson, some of the theorems about triangles have been covered. This lesson will focus on the medians, bisectors, and altitudes of triangles. Additionally, theorems about the intersection points of these segments will be investigated.

Explore

Investigating Medians of a Triangle

Using the following applet, draw the medians of ABC.
Drawing medians of triangle ABC
Do the medians of the triangle intersect in only one point? If yes, calculate the ratio 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.

Concept

Centroid

The centroid of a triangle is the point of intersection of the triangle's medians. The centroid is typically represented by the letter G. This point is always inside the triangle.

Centroid
The centroid of a triangle is also called the center of mass of the triangle.

By means of exploring the applet, it was also seen that the ratio is constant for each median.

Rule

Centroid Theorem

The centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side.

A triangle with its centroid marked.

If AE, BD, and CF are the medians of ABC, then the following statements hold true.

Proof

Consider a triangle with vertices A, B, and C, and two of its medians. Let G be the point of intersection of the medians.

A triangle with two of its medians marked.

Let Q be a point on AC such that QE is parallel to BD.

A triangle with two of its medians marked.

In the diagram, EQC and BDC are corresponding angles. Since EQ and BD are parallel, these two angles are congruent by the Corresponding Angles Theorem. The same is true for QEC and DBC.

A triangle with two of its medians marked.
Therefore, ECQ and BCD have two pairs of congruent angles and are similar by the Angle-Angle Similarity Theorem. Similar reasoning can be used to show that AGD and AEQ are also similar.
By the definition of a median E is the midpoint of BC and therefore E divides BC into two congruent segments. Congruent segments have equal lengths. This information and the Segment Addition Postulate imply that the length of BC is two times the length of EC.
Therefore, the scale factor of the similar triangles is This also means that DC=2QC. By the Segment Addition Postulate DC=DQ+QC. Using the Transitive Property of Equality and the Subtraction Property of Equality the following is obtained.
DC=DQ+QC
2QC=DQ+QC
QC=DQ
Since QC and DQ are equal, Q is the midpoint of DC.
A triangle with two of its medians marked.
Remembering that AD=DC, the ratio of DQ to AD can be calculated.
Corresponding parts of similar triangles are proportional. Since AGD and AEQ are similar, the ratio of DQ to AD is equal to the ratio of GE to AG.
This information can be used to express AG in terms of AE.
AE=AG+GE
Solve for AG
2AE=3AG
This means that AG is two-thirds AE. Now, consider ABC and its medians CF and AE. Let K be the point of intersection of these medians.
A triangle with two of its medians marked.

Let R be a point on AB such that ER is parallel to CF.

A triangle with two of its medians marked.

By following the same reasoning as before, it can be proved that AK is two-thirds AE. Therefore, G and K are the same point which means that the medians are concurrent, or meet at one point.

A triangle with its centroid marked.

Before it was shown that By using similar arguments it can be also shown that and


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.
Drawing a median of the table

Since the centroid of a triangle is the point of intersection of the medians, the centroid will be on this median.

Predicting the centroid of the table

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.

Locating the centroid of the table

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

Centroid of the table


Explore

Investigating Perpendicular Bisectors of a Triangle

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

Concept

Circumcenter

The circumcenter of a triangle is the point of intersection of the triangle's perpendicular bisectors. Circumcenter of a triangle is denoted by the letter S. It can be inside, outside, or on a triangle's side, depending on the triangle type.

Circumcenter

The investigation also indicated that the distances from the circumcenter to each vertex of the triangle are equal.

Rule

Circumcenter Theorem

The circumcenter of a triangle is equidistant from each of the triangle's vertices.

Circumcenter

Based on the characteristics of the diagram, the following relation holds true.

AS=BS=CS

Proof

Assume that ABC is a triangle and DS, ES, and FS are the perpendicular bisectors of the sides of this triangle.

A Triangle with Perpendicular Bisectors

Notice that S is a point on the perpendicular bisector of AB. Therefore, by the Perpendicular Bisector Theorem, S is equidistant from A and B.

Perpendicular Bisector Theorem

Similarly, S is also a point on the perpendicular bisector of BC. Using the Perpendicular Bisector Theorem once again, it can be concluded that S is equidistant from B and C.

Perpendicular Bisector Theorem
By the Transitive Property of Equality, AP is equal to CP.
This proves that AS, BS, and CS are all equal.
Circumcenter Theorem

Two-Column Proof

The proof can be summarized in the following two-column table.

Statements Reasons
Given
Perpendicular Bisector Theorem
AS=CS Transitive Property of Equality

Explore

Investigating Angle Bisectors of a Triangle

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

Concept

Incenter

The incenter of a triangle is the point of intersection of the triangle's angle bisectors. The incenter is typically represented by the letter I. This point is considered to be the center of the triangle. For every triangle, the incenter is always inside the triangle.

Incenter

Rule

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.

Triangle with its incenter marked.

Based on the diagram, the following relation holds true.

DI=EI=FI

Proof

Consider a triangle and its incenter I.

Triangle with its incenter marked.

Let DI, EI, and FI be the distances from I to the sides of the triangle. Recall that the distance from a point to a segment is perpendicular to the segment.

Triangle with its incenter marked.
By the definition of an incenter, AI is the angle bisector of BAC. Since I lies on AI, it is equidistant from the angle's sides by the Angle Bisector Theorem.
Similarly, since I lies on BI, which is the bisector of ABC, it is also equidistant from this angle's sides.
By bringing together the above information, the following is obtained.
This means that I is equidistant from each of the triangle's sides.

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 ABC.
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.


Concept

Orthocenter

The orthocenter of a triangle is the point where a triangle's altitudes intersect. It is usually denoted by the letter H.

Orthocenter of triangles
An acute triangle has its orthocenter inside the triangle. A right triangle's orthocenter is located in the vertex of the right angle. The orthocenter of an obtuse triangle is outside the triangle.

Rule

Orthocenter Theorem

The altitudes of a triangle are concurrent.
Orhtocenter H of a triangle

Proof

Assume that ABC is a triangle with altitudes AK, BL, and CM.

Triangle ABC with its altitudes

Start by drawing lines passing through the vertices A,B, and C, and parallel to the opposite sides BC,AC, and AB, respectively.

Triangle ABC and lines parallel to its sides
Notice that ABCD and AEBC are parallelograms.
Parallelograms ABCD and AEBC
By the Parallelogram Opposite Sides Theorem, BC is congruent to both AD and EA. Since congruent segments have equal lengths, BC is equal to AD and EA.
By the Transitive Property of Equality, AD is equal to EA.
From here, it can be concluded that A is the midpoint of DE.
Midpoint of segment DE

Now, consider the altitude AK from vertex A to the opposite side BC.

Proof

Since AK is perpendicular to BC and BC is parallel to ED, it can be concluded that AK is perpendicular to ED by the Perpendicular Transversal Theorem.

Perpendicular transversal

AK is perpendicular to ED and passes through its midpoint. By the definition of a perpendicular bisector, AK is the perpendicular bisector of ED.

Perpendicular bisector of segment DE

Using this reasoning, it can be proved that the other altitudes of ABC, BL and CM, are perpendicular bisectors of EF and DF, respectively.

Perpendicular Bisectors

By the definition of circumcenter, the perpendicular bisectors AK, BL and CM, intersect at a point H. This implies that the altitudes of ABC are concurrent.

Orthocenter

Two-Column Proof

The proof can be summarized in the following two-column table.

Statements Reasons
ABC is a triangle Given
Given (Drawn)
ABCD,AEBC, and ABFC are parallelograms Definition of a parallelogram
Parallelogram Opposite Sides Theorem
Definition of congruent segments
Transitive Property of Equality
Definition of a midpoint
AK, BL, and CM are the altitudes of ABC Given
Definition of an altitude
Perpendicular Transversal Theorem
Definition of a perpendicular bisector
AK, BL, and CM are concurrent Definition of circumcenter

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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