What Is The Midpoint Theorem?

The midpoint theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length (DE = ½ BC). Its converse states that a line drawn through the midpoint of one side, parallel to another side, bisects the third side.

Imagine sitting in a baseball stadium and watching your favorite team play!! It was a good game, right? Baseball has one of the most uniquely shaped fields in sports, much different than the most common rectangular or circular playing fields, and it is referred to as a Baseball Diamond.

Now, someone with even a bit of mathematical knowledge might wonder how that field was made? Curiosity regarding the construction of the diamond-shaped, field according to the specified dimensions and the theorems involved in its construction, must make one feel inquisitive.

If you take a closer look towards the field or see a birds’ eye view of the field, you will notice that the diamond looks like the sector of a circle with a triangle in it (joining the endpoints of the grass line with the help of a straight line). With that image in mind, let’s explore the mathematics behind the construction of the legendary baseball diamond.

Midpoint Theorem

The midpoint theorem states that “For a given triangle ∆ABC, let D be the midpoint of AB and E be the midpoint of AC. Then the segment DE is parallel to BC, and its length is one half the length of segment BC.”

Or, in plain words, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length. In the figure below, DE ∥ BC and DE = ½ BC.

Midpoint Theorem Proof

Any theorem must have a mathematical proof for it to be valid and the midpoint theorem also has one.

To prove: DE ∥ BC and DE = ½ BC.

In the figure above, extend the line segment DE to a point F such that DE = EF, and join F to point C.

In triangles AED and CEF, we have: AE = CE (since E is the midpoint of AC), ∠AED = ∠CEF (vertically opposite angles), and DE = EF (by construction). So by the SAS criterion, ∆AED ≅ ∆CEF.

Because the triangles are congruent, ∠DAE = ∠FCE and AD = CF. The equal angles ∠DAE and ∠FCE are alternate angles for the lines AB and CF cut by AC, so AB ∥ CF, which means BD ∥ CF.

Now AD = BD (D is the midpoint of AB) and AD = CF, so BD = CF. Quadrilateral BDFC has one pair of opposite sides (BD and CF) that are both equal and parallel, so BDFC is a parallelogram. Hence DF ∥ BC and DF = BC. Since DE = ½ DF, it follows that DE ∥ BC and DE = ½ BC.

Hence, the midpoint theorem is proved.

This is the general textbook explanation that students tend to understand, but never question in terms of its application to real-world problems. Now, before this gets boring, we’ll shift back into baseball to make this concept more interesting and easy to understand.

Practical Understanding

To understand any theorem, it’s essential to understand its practical importance and application. So, we’re coming back to the baseball field for a practical understanding of the theorem. Below are the dimensions of a baseball field (listing only the important/relevant dimensions to prove the practical application of the midpoint theorem).

• Home to first base – 27.43 m
• Third base to home – 27.43 m
• Home plate to left-field foul pole – 99.06 m (325 ft)
• Home plate to right-field foul pole – 99.06 m (325 ft)

The two foul lines meet at home plate at a right angle, so home plate and the two foul poles form a right isosceles triangle. With each equal side measuring 99.06 m, the distance between the two foul poles works out to 99.06 × √2 ≈ 140.09 m. We can now use the midpoint theorem to check this and see whether the theorem holds up in practice.

Considering the triangle formed by the two foul poles and home plate, we have two sides of the triangle, both having their length equal to 99.06 m, and the third side, i.e., the distance between the foul poles, which is 140.09 m.

The midpoints of the equal sides (from home plate to the left and right foul poles) are at a distance of 49.53 m from home and poles. Now, if you join the two midpoints with the help of a line segment, the length of the line segment is unknown, but can be easily determined using basics or trigonometry and triangle congruency.

Here we have A (home plate), B (right foul pole) and C (left foul pole). O is the perpendicular dropped from A to line segment DE.

We will consider AB = AC since in a baseball field, the distance of the two foul poles from home plate is the same. Now, we know that in the triangle AOD, we can calculate DO as follows:

cosine = base / hypotenuse, with ∠ADO = 45° (the field is symmetrical, so the perpendicular AO bisects the apex angle).

Therefore cos 45° = DO / 49.53, which gives the base DO = 49.53 × cos 45° ≈ 35.02 m. Since triangles ADO and AEO are congruent by RHS, O is the midpoint of DE, so DE = 2 × DO ≈ 70.04 m. That is exactly one half of BC (140.09 m), which is just what the midpoint theorem predicts!

Conclusion

If you pay attention, you will see that we are surrounded by real-world examples that can help us learn subjects in a much more unique and fun way. However, it is up to us to find them!

Mathematical theorems have their applications in various fields, but who would have thought that even their favorite sport would have applications from a subject which is a nightmare to most!