🔢 Scale Drawings & Similar Figures
Similar = same shape, different size
▶ Play Door 7Identifying Similar Polygons
Similar figures have the same shape but may differ in size.
Two conditions for similarity:
• All corresponding angles must be EQUAL
• All corresponding sides must be PROPORTIONAL (same scale factor k)
For triangles, if two angles match (AA rule), the triangles are automatically similar.
CheckSimilar?Rectangles 4x6 and 8x12Yes: ratios 8/4=2 and 12/6=2 are equalTriangles 30-60-90 and 30-70-80No: second angles differ (60 vs 70)Any two squaresYes: all angles 90deg, sides always proportional
Worked example
Problem: Are rectangles with sides 4cm x 6cm and 8cm x 12cm similar?
- Ratio of widths: 8/4=2.
- Ratio of lengths: 12/6=2.
- Both ratios are equal, so YES they are similar with scale factor 2.
Answer: Both ratios are equal, so YES they are similar with scale factor 2.
Area Ratio vs Side Length Ratio
When two figures are similar with side length ratio = k, their areas have a squared relationship:
Area Ratio = k²
Similarly, if you know the area ratio, the side length ratio = square root of the area ratio.
Side Ratio (k)Area Ratio (k²)1:21:41:31:92:54:25
Worked example
Problem: Triangle A has side 4cm, similar triangle B has side 12cm. Area of A = 8cm². Find area of B.
- k = 12/4 = 3.
- Area ratio = 3² = 9.
- Area of B = 8 x 9 = 72cm².
Answer: Area of B = 8 x 9 = 72cm².