The converse and its uses

The converse

The basic proportionality theorem starts with parallel lines and ends with equal ratios. Its converse goes the other way: start with equal ratios, end with parallel lines.

Theorem (the converse). If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

If $\dfrac{AD}{DB} = \dfrac{AE}{EC}$, with $D$ on $AB$ and $E$ on $AC$, then $DE \parallel BC$.

Why it is true

We use a proof by contradiction. Suppose $\dfrac{AD}{DB} = \dfrac{AE}{EC}$, but $DE$ is not parallel to $BC$.

1. Draw the line through $D$ that is parallel to $BC$. Let it meet $AC$ at $E'$.
2. By the basic proportionality theorem, $\dfrac{AD}{DB} = \dfrac{AE'}{E'C}$.
3. We were given $\dfrac{AD}{DB} = \dfrac{AE}{EC}$. So $\dfrac{AE'}{E'C} = \dfrac{AE}{EC}$.
4. Add $1$ to both sides. Then $\dfrac{AC}{E'C} = \dfrac{AC}{EC}$, so $E'C = EC$.
5. $E'$ and $E$ lie on $AC$ at the same distance from $C$, so they are the same point.
6. Then $DE$ is the line $DE'$, which is parallel to $BC$. This contradicts what we supposed. So $DE \parallel BC$.

Is DE parallel to BC?

Take $AD = 1.8$ cm, $DB = 1.2$ cm, $AE = 1.5$ cm and $EC = 1$ cm. Then $\dfrac{AD}{DB} = \dfrac{1.8}{1.2} = 1.5$ and $\dfrac{AE}{EC} = \dfrac{1.5}{1} = 1.5$. The ratios are equal, so $DE \parallel BC$.

Now take $AD = 2$ cm, $DB = 3$ cm, $AE = 3$ cm and $EC = 5$ cm. Here $\dfrac{2}{3}$ is not equal to $\dfrac{3}{5}$, so $DE$ is not parallel to $BC$.

A test without fractions: cross-multiply. The ratios are equal exactly when $AD \times EC = DB \times AE$. In the second case $2 \times 5 = 10$ but $3 \times 3 = 9$. They differ.

The midpoint theorem

Let $D$ and $E$ be the midpoints of $AB$ and $AC$. Then $AD = DB$ and $AE = EC$, so $\dfrac{AD}{DB} = 1 = \dfrac{AE}{EC}$. By the converse, $DE \parallel BC$.

This is the midpoint theorem: the line joining the midpoints of two sides of a triangle is parallel to the third side. In the third course of this chapter you will also see that it is half as long as the third side.

The diagonals of a trapezium

In a trapezium $ABCD$ with $AB \parallel DC$, the diagonals $AC$ and $BD$ meet at $O$. Then $\dfrac{AO}{OC} = \dfrac{BO}{OD}$.

1. Draw a line through $O$ parallel to $AB$. It meets $AD$ at $E$. It is also parallel to $DC$.
2. In triangle $ADC$, $EO \parallel DC$. So $\dfrac{AE}{ED} = \dfrac{AO}{OC}$.
3. In triangle $DAB$, $EO \parallel AB$. So $\dfrac{DE}{EA} = \dfrac{DO}{OB}$, which means $\dfrac{AE}{ED} = \dfrac{BO}{OD}$.
4. Both $\dfrac{AO}{OC}$ and $\dfrac{BO}{OD}$ equal $\dfrac{AE}{ED}$. So they are equal.

For example, if $AO = 3$ cm, $OC = 5$ cm and $BO = 4.5$ cm, then $OD = \dfrac{BO \times OC}{AO} = \dfrac{4.5 \times 5}{3} = 7.5$ cm.

Practise with new numbers

Try it in Python

The program uses the converse as a test: it multiplies across and checks whether the two products are equal.

Common mistakes

• Testing only one thing. Equal-looking lengths are not a test. Work out both ratios, or cross-multiply.
• Comparing with a ratio the wrong way round. $\dfrac{AD}{DB}$ goes with $\dfrac{AE}{EC}$, not with $\dfrac{EC}{AE}$.
• Using the converse for points that are not on the sides. $D$ must lie on $AB$ and $E$ on $AC$.
• Pairing the wrong parts of a trapezium. $AO$ goes with $OC$ on one side, and $BO$ goes with $OD$ on the other.

Remember these

• If $\dfrac{AD}{DB} = \dfrac{AE}{EC}$, then $DE \parallel BC$. This is the converse.
• To test for parallel lines, cross-multiply: $AD \times EC = DB \times AE$.
• The line joining the midpoints of two sides is parallel to the third side.
• In a trapezium, $\dfrac{AO}{OC} = \dfrac{BO}{OD}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. $D$ is on $AB$ and $E$ is on $AC$. $AD = 4$ cm, $DB = 6$ cm, $AE = 3$ cm and $EC = 4.5$ cm. Is $DE$ parallel to $BC$?

2. Now take $AD = 6$ cm, $DB = 9$ cm, $AE = 8$ cm and $EC = 10$ cm. Is $DE$ parallel to $BC$?

3. In your own words, what does the midpoint theorem say?

4. In a trapezium $ABCD$ with $AB \parallel DC$, the diagonals meet at $O$. $AO = 2$ cm, $OC = 6$ cm and $BO = 3$ cm. Find $OD$.

5. Write the converse of the basic proportionality theorem in one sentence.