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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.

Rule

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.

Proof

Triangle Midsegment Theorem


If is a midsegment of and

Proof

Draw an arbitrary triangle, in the coordinate plane so that lies on the origin, and is horizontal and lies on the -axis.

Since lies on the origin, its coordinates are Point is on the -axis, meaning its -coordinate is The remaining coordinates are unknown, and can be named and Then, the points are Draw midsegment from to By defintion, is the midpoint of and the midpoint of

Proof

parallel to

If the slopes of and are equal, the segments are parallel. Since was drawn as a horizontal line, its slope equals To show that the slope of also equals it can be shown that the -coordinates of its points are the same. To do this, and and the midpoint formula can be used. The -coordinates of and are both It follows that the slope of equals Thus, since and have equal slopes, they are parallel.

Proof

is half the length of

Since both and are horizontal, their lengths are given by the difference of the endpoints' -values. The -coordinates of and are and respectively. This gives To find the -coordinates of and the midpoint formula is used. The length of is given by subtracting the -coordinates: Since is half of the midsegment is half the length of

Therefore, a midsegment of a triangle is parallel to the third side of the triangle and half the length.
Rule

Triangle Proportionality Theorem

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

For since

This can be proven using congruent angles.

Proof

Triangle Proportionality Theorem


Since and are parallel, the corresponding angles and are congruent. Similarly, and are congruent. Also, triangles and share the angle

This means that and are the same shape, but is a smaller version. Thus, the ratios between corresponding sides of the triangles are equal. This leads to Using that and it's possible to rearrange this equality to get the proportionality theorem.
Thus, the following equality yields Therefore, a segment parallel to one side of a triangle divides the other two sides proportionally. This can be summarized in a flowchart proof.
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.


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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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