Similar Triangles

Worked example

1. 1

   $A\leftrightarrow P,B\leftrightarrow R,C\leftrightarrow Q$ — Pair the corners that have equal angles. $A$ goes with $P$ and $B$ with $R$, so the two left over, $C$ and $Q$, go together.

2. 2

   $AB\leftrightarrow PR,AC\leftrightarrow PQ$ — A side joins two corners, so it matches the side that joins their partners. $AB$ joins $A$ and $B$, so it matches $PR$. $AC$ joins $A$ and $C$, so it matches $PQ$.

3. 3

   $AC\leftrightarrow PQ:6\to 9$ — Choose a matching pair where you know both lengths: 6 cm and 9 cm.

4. 4

   $k=9÷6=1.5$ — The scale factor is big over small, because the 6 has to grow into the 9.

5. 5

   $PR=AB\times 1.5$ — $PR$ is the side you want. It matches $AB$, not $PQ$, and it is on the bigger triangle, so multiply by the scale factor.

6. 6

   $PR=8\times 1.5=12\text{ cm}$ — Work it out. 12 is bigger than 8, as it should be for the bigger triangle.

Worked example

1. 1

   $AB=AD+DB=4+6=10$ — The big triangle's side is the whole line $AB$, not just the part $DB$. Add the two parts.

2. 2

   $AD\leftrightarrow AB:4\to 10$ — $AD$ is the small triangle's side and $AB$ is the matching side of the big triangle. Both lengths are known.

3. 3

   $k=10÷4=2.5$ — The scale factor is big over small.

4. 4

   $BC=DE\times 2.5$ — $DE$ is parallel to $BC$, so they match. $BC$ is on the big triangle, so multiply by the scale factor.

5. 5

   $BC=5\times 2.5=12.5\text{ cm}$ — Work it out. 12.5 is bigger than 5, as it should be.

Worked example

1. 1

   $\text{angle }ACB=\text{angle }ECD$ — These angles are vertically opposite, so they are equal.

2. 2

   $\text{angle }A=\text{angle }E$ — $AB$ is parallel to $DE$, so the alternate angles are equal. $A$ goes with $E$, so $B$ goes with $D$.

3. 3

   $AB\leftrightarrow ED:6\to 15$ — Both sides are opposite the angle at $C$, so they match. Both lengths are known.

4. 4

   $k=15÷6=2.5$ — The scale factor is big over small.

5. 5

   $CE=AC\times 2.5$ — $A$ matches $E$, so $AC$ matches $EC$. $CE$ is on the bigger triangle, so multiply.

6. 6

   $CE=4\times 2.5=10\text{ cm}$ — Work it out. This is the side you wanted.

7. 7

   $AE=AC+CE=4+10=14\text{ cm}$ — Only if they ask for the whole line $AE$: it runs through $C$, so add the two parts.

Worked example

1. 1

   $\frac{AB}{AD}=\frac{AC}{AE}$ — $AD$ and $AE$ are sides of the small triangle. $AB$ and $AC$ are their matches on the big one. Big over small gives the same scale factor each time, so the two fractions are equal.

2. 2

   $AB=x+3,AC=4+2=6$ — The big triangle's sides are the whole lines, so add the parts.

3. 3

   $\frac{x+3}{x}=\frac{6}{4}$ — Put the lengths into the equation.

4. 4

   $4(x+3)=6x$ — Cross multiply: multiply both sides by 4 and by $x$ to clear the fractions.

5. 5

   $4x+12=6x$ — Expand the bracket: $4\times x$ and $4\times 3$.

6. 6

   $12=2x$ — Take $4x$ from both sides, so the $x$ terms are together.

7. 7

   $x=6$ — Divide both sides by 2.

8. 8

   $\frac{6+3}{6}=1.5=\frac{6}{4}$ — Check: both scale factors come out as 1.5.

Worked example

1. 1

   $4,6,7\text{and}10,15,17.5$ — List each triangle's sides from shortest to longest. The shortest sides match, the middle ones match and the longest ones match.

2. 2

   $\frac{10}{4}=2.5$ — Shortest sides: big over small.

3. 3

   $\frac{15}{6}=2.5$ — Middle sides: the same division.

4. 4

   $\frac{17.5}{7}=2.5$ — Longest sides: the same division.

5. 5

   $\text{all three ratios equal}$ — Every pair gives 2.5, so the triangles are similar with scale factor 2.5. Write this sentence in your answer.