Triangle Midsegment Theorem

There are so many rules regarding triangles to learn about. Once we understand these rules, we can solve triangles and many other geometrical problems with a few easy steps. One of the most useful tricks is the triangle midsegment theorem. But wait a second.. What exactly is a midsegment? How does this theorem work? Let's find out:

The triangle midsegment theorem, explained

In the world of math, we can rely on theorems because other mathematicians have already proven that they are correct. Once proven, theorems become rules that we can apply to many different situations.

The triangle midsegment theorem states that:

• A midsegment connecting two sides of a triangle is parallel to the third side

And:

• The midsegment is half as long as the third side

Remember that a midsegment is simply a line that connects the midpoints of two triangle sides. You might also remember that a midpoint is the halfway point of each triangle side. We must draw our midsegments between these two midpoints. This is what a midsegment looks like:

As we can see, points D and E are both midpoints, and they are both halfway between points A-B and A-C, respectively. We can call this midsegment DE.

Visualizing the triangle midsegment theorem

Let''s take a closer look at our midsegment:

We can see that both distances on either side of D and E are equal. If we know this is true, then we can apply the midsegment triangle theorem. We can also put this theorem into a handy formula:

If $\mathrm{AD}=\mathrm{DB}$ and $\mathrm{AE}=\mathrm{EC}$, then $\mathrm{DE}\parallel \mathrm{BC}$ and $\mathrm{DE}=\genfrac{}{}{0ex}{}{1}{2}\mathrm{BC}$. Note that ∥ is the symbol for parallelism.

Using the triangle midsegment theorem to solve problems

Now let's put our knowledge of the triangle midsegment theorem to good use. Consider the following diagram:

We can see that P and Q are the midpoints of sides AB and AC, respectively. We know this because the values on each side of P and Q are equal. This lets us know that PQ is the midsegment with an unknown length (x).

We also know that line BC has a value of 6.

What is the length of the midsegment in this diagram? Finding the answer is simple if we apply the triangle midsegment theorem:

$\mathrm{PQ}=\genfrac{}{}{0ex}{}{1}{2}\mathrm{BC}$

In other words, all we need to do is divide line BC by 2: $\genfrac{}{}{0ex}{}{1}{2}\left(6\right)=3$

The midsegment is exactly 3 units in length.

How do we know that the triangle midsegment theorem is correct?

Whenever we face a difficult geometrical exercise, it always helps to construct triangles within other shapes to find our answers.

We can prove the validity of the triangle midsegment theorem with the same strategy. If we construct a triangle with vertices of all three midpoints of a triangle, we are left with one triangle in the middle. We also get three triangles surrounding this central triangle.

Try it yourself. Even with a quick glance, you will see that the midsegment is equal to two halves of the third side.

Topics related to the Triangle Midsegment Theorem

Triangles

30-60-90 Triangles

Triangle Angle Bisector Theorem

Flashcards covering the Triangle Midsegment Theorem

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Triangle Midsegment Theorem

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

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