What is the relation between the line segment joining the midpoint and two sides and third side of a triangle?

This is not quite right. Firstly, as an aside, we don't talk about "lengths of points". As such, your first statement about the length of $A$ doesn't quite make sense.

You then make a statement about the line $DE$. You say that it's "half-way between" something and something else. What is the meaning of this? What does it mean for a whole line to be halfway between something and something else? If $DE$ was not a straight line, would this statement still stand? What if $DE$ was "rotated" a little?

Certainly your final assertion isn't right. What is the logical step between $DE$ being halfway between $A$ and $BC$, and $DE$ having half the length of $BC$? This does not follow in any simple way.

As such, your proof has a few holes, and I suspect is not quite going to be fruitful enough to prove your statement. I hope this helps.

In geometry, the mid-point theorem helps us to find the missing values of the sides of the triangles. It establishes a relation between the sides of a triangle and the line segment drawn from the midpoints of any two sides of the triangle. The midpoint theorem states that the line segment drawn from the midpoint of any two sides of the triangle is parallel to the third side and is half of the length of the third side of the triangle.

In this article, we will explore the concept of the midpoint theorem and its converse. We will learn the application of the theorem with the help of a few solved examples for a better understanding of the concept.

What is Midpoint Theorem?

The midpoint theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of the length of the third side. This theorem is used in various places in real life, for example in the absence of a measuring instrument, we can use the midpoint theorem to cut a stick into half.

Midpoint Theorem Definition

The midpoint theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of the third side. Consider an arbitrary triangle, ΔABC. Let D and E be the midpoints of AB and AC respectively. Suppose that you join D to E. The midpoint theorem says that DE will be parallel to BC and equal to exactly half of BC. Look at the image given below to understand the triangle midpoint theorem.

Midpoint Theorem Proof

Now, let us state and prove the midpoint theorem. The straight line joining the midpoints of any two sides of the triangle is considered parallel and half of the length of the third side. Consider the triangle ABC, as shown in the figure below. Let E and D be the midpoints of the sides AC and AB respectively. Then the line DE is said to be parallel to the side BC, whereas the side DE is half of the side BC, i.e.

DE || BC

DE = 1/2 × BC

This is the statement of the midpoint theorem. Now let us look at its proof.

Given: D and E are the mid-points of sides AB and AC of ΔABC respectively.

Construction: In ΔABC, through C, draw a line parallel to BA, and extend DE such that it meets this parallel line at F, as shown below:

Proof: Compare ΔAED with ΔCEF:

By the ASA criterion, the two triangles are congruent. Thus, DE = EF and AD = CF. But AD is also equal to BD, which means that BD = CF (also, BD || CF by our construction). This implies that BCFD is a parallelogram. Thus,

DF || BC ⇒ DE || BC

and, DF = BC

⇒ DE + EF = BC

⇒ 2DE = BC (as, DE = EF, proved above)

⇒ DE = 1/2 × BC

This completes our proof. Will the converse of the midpoint theorem hold? Yes, it will, and the proof of the converse is presented next.

Converse of Midpoint Theorem

The midpoint theorem converse states that the line drawn through the midpoint of one side of a triangle that is parallel to another side will bisect the third side. Consider a triangle ABC, and let D be the midpoint of AB. A line through D parallel to BC meets AC at E, as shown below. Now suppose that E is not the midpoint of AC. Let F be the midpoint of AC. Join D to F, as shown below:

By the midpoint theorem, DF || BC. But we also have DE || BC. This cannot happen because through a given point (in this case, D), exactly one parallel can be drawn to a given line (in this case, BC). Thus, E must be the midpoint of AC. This completes our proof of the converse midpoint theorem.

Midpoint Theorem Formula

In math, we also have a midpoint theorem formula which has its applications in coordinate geometry. It can also be known as the midpoint theorem of a line segment. It states that if we have a line segment whose endpoints coordinates are given as (x1, y1) and (x2, y2), then we can find the coordinates of the midpoint of the line segment by using the formula given below:

Let (xm, ym) be the coordinates of the midpoint of the line segment. Then,

(xm, ym) = ( (x1 + x2)/2 , (y1 + y2)/2 )

This is known as the midpoint theorem formula.

Sides Joining the Midpoints of a Triangle

An interesting consequence of the midpoint theorem is that if we join the midpoints of the three sides of any triangle, we will get four (smaller) congruent triangles, as shown in the figure below:

We have: ΔADE ≅ ΔFED ≅ ΔBDF ≅ ΔEFC.

Proof: Consider the quadrilateral DEFB. By the midpoint theorem, we have:

• DE = 1/2 × BC = BF
• DE || BF

Thus, DEFB is a parallelogram, which means that ΔFED ≅ ΔBDF. Similarly, we can show that AEFD and DECF are parallelograms, and hence all the four triangles so formed are congruent to each other (make sure that when you write the congruence relation between these triangles, you get the order of the vertices correct).

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Important Notes on Midpoint Theorem

• The midpoint theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of the length of the third side.
• The midpoint theorem converse states that the line drawn through the midpoint of one side of a triangle that is parallel to another side will bisect the third side.

1. Example 1: Consider a triangle ABC, and let D be any point on BC. Let X and Y be the midpoints of AB and AC respectively.

   

   Show that XY bisects AD.

   Solution: It is given that X and Y are the midpoints of AB and AC. By the midpoint theorem, XY || BC. Now, consider ΔABD. The segment XE is parallel to the base BD, and X is the midpoint of AB. By the converse of the midpoint theorem, E must be the midpoint of AD. Thus, XY bisects AD.

2. Example 2: Prove that if three parallel lines make equal intercepts on one transversal, then they will make equal intercepts on any other transversal as well.

   Solution: Let us first understand this problem in a better way. Consider three lines and two transversals, as shown below:

   

   Suppose that the intercepts on the left transversal are equal, that is, AB = BC. We then have to prove that the intercepts on the right transversal will also be equal, that is, DE = EF.

   To prove this, join A to F:

   

   Consider ΔACF. Since B is the midpoint of AC and BG || CF, the converse of the midpoint theorem tells us that G is the midpoint of AF. Now, consider ΔAFD. We have shown that G is the midpoint of AF. Also, GE || AD. Thus, the converse of the midpoint theorem tells us that E must be the midpoint of FD. Therefore, DE = EF.

3. Example 3: Consider a parallelogram ABCD. E and F are the midpoints of AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.

   Solution: Consider the following figure:

   

   We have to show that BX = XY = YD = BD/3.

   First of all, we note that AECF is a parallelogram as ABCD is a parallelogram which means AB = CD and so AE = CF (as E and F are the mid-points), and thus, EC || AF. Now, consider ΔBAY. Since E is the midpoint of AB, and EX || AY, the converse of the midpoint theorem tells us that X is the midpoint of BY, which means that BX = XY. Similarly, we can prove that XY = YD. Thus, BX = XY = YD = BD/3. Hence proved.

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FAQs on Midpoint Theorem

The midpoint theorem states that in any triangle, the line joining the mid-points of any two sides of the triangle is parallel to and half of the length of the third side. It has many applications in math while calculating the sides of the triangle, finding the coordinates of the mid-points, etc.

How do you Prove Mid Point Theorem?

To prove the midpoint theorem, we use the congruency rules. We construct a triangle outside the given triangle such that it touches the side of the triangle. And then we prove that it is congruent to any one part of the triangle. It helps us to prove the equality between sides by using CPCTC rules.

How to Prove Converse Mid Point Theorem?

To prove the converse of the midpoint theorem, consider a triangle ABC, and let D be the midpoint of AB. A line through D parallel to BC meets AC at E. Now suppose that E is not the midpoint of AC. Let F be the midpoint of AC. Join D to F. By the midpoint theorem, DF || BC. But we also have DE || BC. This cannot happen because through a given point (in this case, D), exactly one parallel can be drawn to a given line (in this case, BC). Thus, E has to be the midpoint of AC. This is the proof of the converse of the midpoint theorem.

How do you Find the Midpoint Theorem?

The midpoint theorem can be applied to any triangle. When a line is drawn between the midpoints of any two sides of the triangle, it is always parallel to and half of the length of the third side. This theorem is applicable in all types of triangles.

What is the Statement of Midpoint Theorem?

The midpoint theorem statement is that "A line drawn between the midpoints of any two sides of a triangle is parallel to and half of the third side of the triangle". It can be mathematically represented as,

Suppose DE is the line joining the mid-points of triangle ABC and it is parallel to BC.

⇒ DE || BC and DE = 1/2 × BC

Where is the Midpoint Theorem Used?

The midpoint theorem is used to define the relationships between the sides of the triangle. It is useful to find the missing side lengths, to prove the congruency of four triangles formed by joining the mid-points of the triangle, to find coordinates, etc. All these are the applications of the mid-point theorem in math.