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Midpoint Theorem - Statement, Proof, and Examples

TL;DR

The midpoint theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and exactly half its length. This article gives the formal statement, a full labelled proof, the converse, six worked examples, and the common mistakes, plus how the theorem relates to the triangle's midsegment.

A Shortcut That Measures A Distance You Can't Reach

Suppose you need the width of a river but can only walk one bank. Mark the midpoints of two survey lines running back from the water, join them, measure that short segment, and double it - you now have the width without crossing. That trick is the midpoint theorem in action: join two midpoints, and the segment you draw is always exactly half the far side, and always parallel to it.

What Does The Midpoint Theorem State?

The midpoint theorem states: in any triangle, the line segment joining the midpoints of two sides is parallel to the third side and equal to half its length. Formally, in △ABC, if D is the midpoint of AB and E is the midpoint of AC, then DE∥BC and DE=1/2 BC.

A midpoint is the point that divides a segment into two equal halves; the midpoint formula locates it from coordinates. The theorem delivers two conclusions at once, and both matter: a direction result (the new segment runs parallel to the third side) and a length result (it is exactly half as long). Missing either half means missing the theorem.

The segment DE itself has a name: it is a midsegment of the triangle. This page focuses on the theorem - its statement and proof - while the companion page on the midsegment of a triangle focuses on the segment and its properties. They describe the same figure from two angles: the theorem is the rule, the midsegment is the object the rule is about.

How Is The Midpoint Theorem Proved?

The proof is worth doing once, because seeing why DE is half of BC makes the result stick far better than memorising it. The strategy: extend DE to build a parallelogram, then let the parallelogram's own properties finish the job. The labelled diagram below tracks every point the proof names.

Given: In △ABC, D and E are the midpoints of AB and AC, so AD=DB and AE=EC.

To prove: DE∥BC and DE=1/2 BC.

Construction: Extend DE to a point F such that EF=DE. Join C.

Proof, step by step:

In △AED and △CEF:

AE=EC ∠AED=∠CEF (vertically opposite angles) DE=EF (by construction)

So △AED≅△CEF by the SAS congruence rule.

Therefore AD=CF and ∠ADE=∠CFE (corresponding parts of congruent triangles).

Because ∠ADE=∠CFE are equal alternate angles, AD∥CF, which means DB∥CF.

Now AD=DB (D is a midpoint) and AD=CF (just shown), so DB=CF.

Since DB∥CF and DB=CF, the quadrilateral BCFDB is a parallelogram.

In a parallelogram opposite sides are equal and parallel, so:

DF∥BC, hence DE∥BC.

DF=BC.

But DF=DE+EF=2DE, so 2DE=BC, giving DE=1/2 BC.

Both conclusions are proved: DE∥BC and DE=1/2 BC.

What Is The Converse Of The Midpoint Theorem?

The converse runs the theorem backward: the line drawn through the midpoint of one side of a triangle, parallel to a second side, bisects the third side. In △ABC, if D is the midpoint of AB and a line through D is parallel to BC, that line meets AC at its midpoint E.

A compact way to hold both directions:

• Midpoint theorem: midpoint + midpoint → parallel and half.
• Converse: midpoint + parallel → the other midpoint.

The converse is what lets you prove a point is a midpoint using only a parallel line - no measuring required. It is the tool behind many constructions and coordinate-geometry proofs involving similar triangles.

Examples of the Midpoint Theorem

Six examples, from a one-step length to a full quadrilateral argument.

Example 1

In △ABC, D and E are midpoints of AB and AC. If BC = 10 cm, find DE.

By the midpoint theorem, DE=1/2 BC.

DE=1/2 × 10

DE=5 cm

Final answer: 5 cm.

Example 2

In △PQR, M and N are midpoints of PQ and PR. A student measures MN = 6 cm and concludes QR = 6 cm. What went wrong?

The intuitive move is to treat the midsegment and the third side as equal since both look like "the bottom" of a smaller and larger triangle sitting together.

That skips the halving. The theorem says the midsegment is half the third side, not equal to it. Reading MN=QR drops the factor of 1/2.

The correct method doubles the midsegment to recover the third side:

MN=1/2 QR

QR=2×MN=12 cm

Final answer: 12 cm. The midsegment is the half, so the third side is the double - never equal.

Example 3

In △ABC, DE joins midpoints of AB and AC with DE = 4.5 cm. Find BC.

DE=1/2 BC, so BC=2×DE.

BC=2×4.5

BC=9 cm

Final answer: 9 cm.

Example 4

In △ABC, D is the midpoint of AB. A line through D parallel to BC meets AC at E. If AC = 14 cm, find AE.

This uses the converse: a line through one midpoint, parallel to a second side, bisects the third side. So E is the midpoint of AC.

AE=1/2 AC

AE=7 cm

Final answer: 7 cm.

Example 5

The sides of △ABC are 12 cm, 16 cm, and 20 cm. Find the perimeter of the triangle formed by joining the midpoints of its three sides.

Each of the three midsegments is half the side it is parallel to. So the midpoint triangle's sides are:

• 6 cm
• 8 cm
• 10 cm

Perimeter =6+8+10=24 cm.

Final answer: 24 cm — exactly half the original triangle's perimeter of 48 cm.

Example 6

In quadrilateral ABCD, P, Q, R, S are the midpoints of sides AB, BC, CD, DA. Show that PQRS is a parallelogram.

Draw the diagonal AC. In △ABC, P and Q are midpoints of AB and BC, so by the midpoint theorem PQ∥AC and PQ=1/2 AC.

In △ADC, S and R are midpoints of AD and CD, so SR∥AC and SR=1/2 AC.

Therefore PQ∥SR and PQ=SR.

Final answer: One pair of opposite sides of PQRS is equal and parallel, so PQRS is a parallelogram. This result, called Varignon's theorem, falls straight out of two applications of the midpoint theorem.

Why The Midpoint Theorem Matters: Halving Without Measuring

The theorem's power is that it turns a hard-to-reach length into an easy one, and it does so without any measuring tool touching the far side.

• It computes inaccessible distances. Surveyors and navigators use midpoint reasoning to find a distance across water or terrain they cannot cross, then double the accessible half.
• It underpins the coordinate midpoint. The coordinate midpoint formula is the algebraic face of this same halving idea, letting the theorem run inside coordinate proofs.
• It scales triangles cleanly. The midpoint triangle is a half-size copy of the original.

This halving-and-parallel structure is the seed of similar-triangle theory, which is why the midpoint theorem is usually a student's first proof that connects equal division to parallelism.

What Are The Most Common Mistakes With The Midpoint Theorem?

Two mistakes cause most wrong answers, and both come from mishandling the factor of one half.

Mistake 1: Forgetting the factor of one half

Where it slips in: When the midsegment length is given and the third side is wanted, the halving is easy to reverse the wrong way - or drop entirely.

Don't do this: Setting the midsegment equal to the third side.

The correct way: The midsegment is the half; the third side is the double. Going from midsegment to third side, multiply by 2.

Mistake 2: Applying the theorem when the points are not midpoints

Where it slips in: When a segment joins two points on the sides that merely look central but are not stated to be midpoints.

Don't do this: Using DE=1/2 BC for a segment DE whose endpoints are not confirmed midpoints.

The correct way: The theorem needs both endpoints to be true midpoints. If only one is, use the converse instead.

Conclusion

• The midpoint theorem says the segment joining two midpoints of a triangle is parallel to the third side and half its length.
• The proof extends the midsegment to build a parallelogram, then reads off both results from its properties.
• The converse reverses it: a midpoint plus a parallel line locates the second midpoint.
• The segment involved is the triangle's midsegment.

Practice These To Solidify Your Understanding

Work through these, then check:

1. In △ABC, D and E are midpoints of AB and AC. If BC = 18 cm, find DE. (Answer to Question 1: 9 cm.)
2. A midsegment measures 7.5 cm. Find the third side it is parallel to. (Answer to Question 2: 15 cm.)
3. A triangle has sides 8, 10, and 14 cm. Find the perimeter of its midpoint triangle. (Answer to Question 3: 16 cm.)