Similar Triangles and the Basic Proportionality Theorem: Quick Revision

A brief note on Similar Triangles and the Basic Proportionality Theorem. Read it before a test. Try each question at the end before you open its answer.

Similar figures

• Matching angles are equal, and matching sides are in one ratio, the scale factor.
• Any two circles, any two squares and any two equilateral triangles are similar.
• Write $\triangle ABC \sim \triangle DEF$ with the vertices in matching order.
• To find a side: match shortest to shortest, find the scale factor, and multiply.

The basic proportionality theorem

• If $DE \parallel BC$, with $D$ on $AB$ and $E$ on $AC$, then $\dfrac{AD}{DB} = \dfrac{AE}{EC}$.
• The same fact: $\dfrac{AB}{DB} = \dfrac{AC}{EC}$ and $\dfrac{AD}{AB} = \dfrac{AE}{AC}$.
• The proof compares areas of triangles with the same height, and uses the parallel lines to show that $\text{ar}(BDE) = \text{ar}(CDE)$.

The converse and its uses

• If $\dfrac{AD}{DB} = \dfrac{AE}{EC}$, then $DE \parallel BC$. Test by cross-multiplying: $AD \times EC = DB \times AE$.
• Midpoint theorem: the line joining the midpoints of two sides is parallel to the third side.
• In a trapezium with $AB \parallel DC$, the diagonals cut each other so that $\dfrac{AO}{OC} = \dfrac{BO}{OD}$.

A worked example

In triangle $ABC$, $DE \parallel BC$, with $AD = 6$ cm, $DB = 9$ cm and $AC = 25$ cm. Find $AE$.

1. The ratio. $\dfrac{AD}{DB} = \dfrac{6}{9} = \dfrac{2}{3}$, so $AE : EC = 2 : 3$.
2. The whole. $AC$ is $5$ parts, so one part is $25 \div 5 = 5$ cm.
3. The answer. $AE = 2 \times 5 = 10$ cm.

Check. $EC = 15$ cm, and $\dfrac{10}{15} = \dfrac{2}{3}$.

Practice with new numbers

A program to try

The program finds the missing length $EC$ for three triangles, and checks that both ratios agree.

Mistakes to avoid

• Pairing a piece with the whole. $\dfrac{AD}{DB}$ goes with $\dfrac{AE}{EC}$.
• Using the theorem without parallel lines.
• Turning only one ratio upside down.
• Testing parallel lines by eye. Work out both ratios.

Check yourself

1. In triangle $ABC$, $DE \parallel BC$, with $AD = 8$ cm, $DB = 12$ cm and $AE = 10$ cm. Find $EC$.

2. Is $DE \parallel BC$ when $AD = 3$ cm, $DB = 5$ cm, $AE = 6$ cm and $EC = 9$ cm?

3. In a trapezium $ABCD$ with $AB \parallel DC$, the diagonals meet at $O$. $AO = 4$ cm, $OC = 10$ cm and $BO = 6$ cm. Find $OD$.

4. Triangle $ABC$ is similar to triangle $DEF$, with $AB = 3$ cm and $DE = 12$ cm. What is the scale factor?