Similarity and Lengths

Similarity and Lengths

Two shapes are similar if they have the same shape — all corresponding angles are equal and all corresponding lengths are in the same ratio (the scale factor \( k \)). Similarity is a proportional relationship: every length in the larger shape is \( k \) times the corresponding length in the smaller.

Finding and Using the Scale Factor

\[ k = \frac{\text{image length}}{\text{original length}} \]

Once \( k \) is found from one pair of corresponding lengths, all other corresponding lengths can be calculated by multiplying (original → image) or dividing (image → original).

Proving Triangles are Similar

Two triangles are similar if any one of the following holds:

  • AA (Angle-Angle): two pairs of corresponding angles are equal (the third follows automatically).
  • SAS similarity: two pairs of sides are in the same ratio and the included angle is equal.
  • SSS similarity: all three pairs of sides are in the same ratio.

AA is the most commonly used criterion at GCSE. Once similarity is established, all corresponding lengths are in the ratio \( k : 1 \).

Similar Triangles in Diagrams

Similar triangles often appear within diagrams — e.g. a triangle inside a larger triangle, or formed by parallel lines. Identify corresponding vertices carefully before setting up the ratio.

Worked Examples

Two similar triangles have corresponding sides 6 cm and 10 cm. A second side of the smaller triangle is 9 cm. Find the corresponding side of the larger.
\[ k = \frac{10}{6} = \frac{5}{3} \qquad \text{larger side} = 9\times\frac{5}{3} = 15 \text{ cm} \]
Triangles \( ABC \) and \( PQR \) have \( \angle A=\angle P=55° \) and \( \angle B=\angle Q=70° \). Prove they are similar and state the scale factor if \( AB=8 \) cm and \( PQ=12 \) cm.

\( \angle A=\angle P \) and \( \angle B=\angle Q \) (given). Therefore \( \angle C=\angle R=55° \) (angle sum). Triangles are similar by AA.

Scale factor: \( k=12/8=3/2 \).

In triangle \( ABD \), a line \( CE \) is drawn parallel to \( BD \) with \( C \) on \( AB \) and \( E \) on \( AD \). If \( AC=4 \), \( AB=10 \), find \( CE:BD \).

\( CE \parallel BD \implies \angle ACE=\angle ABD \) (corresponding). \( \angle A \) is common. So \( \triangle ACE \sim \triangle ABD \) (AA).

\[ k=\frac{AC}{AB}=\frac{4}{10}=\frac{2}{5} \implies CE:BD=2:5 \]

 Key Takeaways

  • Similar shapes: equal angles, proportional sides. Scale factor \( k=\) image ÷ original.
  • AA criterion: two equal angles → similar triangles (used most frequently).
  • Parallel lines create similar triangles — use corresponding and alternate angle properties to identify equal angles.
  • Always identify corresponding vertices correctly before forming the ratio.
  • Once \( k \) is known: all lengths in the image = \( k \times \) corresponding lengths in the original.