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Triangles

Dissecting Triangles

Every triangle possesses characteristics that can be used to further analyze the triangle. These include medians, midsegments, and the centroid.
Rule

Perpendicular Bisector Theorem

Any point on a perpendicular bisector is equidistant from the endpoints of the line segment.

This can be proven using congruent triangles.

Proof

Perpendicular Bisector Theorem


Suppose is the perpendicular bisector of and that is the midpoint of

Two triangles can be created by connecting points and and and

These triangles both have a right angle and one of the legs measures half of They also share one leg,

According to the SAS Congruence Theorem, the triangles are congruent. Thus, their hypotenuses are also congruent.

Therefore, any point on a perpendicular bisector is equidistant from the endpoints of the segment. This can be summarized in a two-column proof.

Statement Reason
Given
SAS congruence theorem
Definition of congruent segments
Note that and are not triangles if is the point of intersection, However, since is the midpoint of it is, by definition, equidistant from and
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Exercise

Determine the perimeter of

Show Solution
Solution
To determine the perimeter, we need the lengths of each side of the triangle. The line divides in two equal parts at a right angle, meaning it's a perpendicular bisector. According to the Perpendicular Bisector Theorem, the segments and are congruent. Therefore, Let's solve for
Since is units long. The segment is congruent to which means that in total measures units. We can use to find the lengths of Since also measures units. We can now add all three side lengths to determine the perimeter. The perimeter of the triangle is units.
Concept

Midsegment

A midsegment is a segment that connects the midpoints of two of the sides of a triangle.

Since a triangle has three sides, there are three midsegments.

Proof

Triangle Midsegment Theorem

The segment that connects the midpoints of two sides of a triangle — a midsegment — is parallel to the third side of the triangle and half the length.

If is a midsegment of and

This can be proven in the coordinate plane.
Rule

Triangle Proportionality Theorem

If a segment parallel to one of the sides of a triangle is drawn between the other sides, the segment divides the other two sides proportionally.

Triangle ABC with segment DE drawn parallel to AB

Based on the diagram, the following relation holds true.

If then

Proof

Since and are parallel, by the Corresponding Angles Theorem, and are congruent. Similarly, and are congruent.

Triangle ABC with segment DE drawn parallel to AB
Therefore, by the Angle-Angle Similarity Theorem, and are similar. Consequently, their corresponding sides are proportional. Applying the Segment Addition Postulate, both numerators can be rewritten. Substituting these expressions into the equation above, the required proportion will be obtained.
Concept

Median

A median of a triangle is a line segment between the midpoint of one side and the opposite vertex.

Since a triangle has three vertices, every triangle has three medians.

All medians intersect each other at one point, the centroid.
Concept

Centroid

The three medians of a triangle intersect at one point called the centroid.

Rule

Centroid Theorem

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

If and are the medians of then

This can be proven using midpoints and parallel lines.

Proof

Centroid Theorem


Consider The points and are midpoints on their respective side. Thus, and are medians.

The two medians intersect at the point

Now, two new points are introduced — the midpoints of and Call them and

Since and are midpoints of and is a midsegment of Thus, by the Triangle Midsegment Theorem, is parallel to and half the length of

Similarly, is a midsegment of since and are the midpoints of and Therefore, is also parallel to and half the length of It follows that

Since and are parallel and congruent, are the vertices of a parallelogram. By the Parallelogram Diagonals Theorem, the diagonals of a parallelogram bisect each other. Therefore, and

Thus, the median intersects at two-thirds of the distance from Now, by applying the same reasoning for the third median, it also intersects at two-thirds from

The median from point also intersects at two-thirds from point Therefore, the centroid of a triangle is two-thirds the distance from each vertex to the midpoint of the opposite side.


fullscreen
Exercise

Determine the length of

Show Solution
Solution
To begin, notice that the segments inside the triangle each bisect their respective sides. Thus, they are medians and intersect at the centroid. By the Centroid Theorem, the sides that measures is twice as long as the segment that measures We can write the following equation. Solving for will allow us to determine the length of the smaller segments, which we can then use to find the length of
Now we can use the value of to find the length of The length of is units.
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