Similar Triangles: Conditions, BPT and Area Ratios
What the exam tests
Similar Triangles questions test the AAA/AA, SSS and SAS similarity conditions, the Basic Proportionality Theorem, and the ratios of sides, perimeters and areas. You must match corresponding vertices before writing proportions.
Core rules
- Similar triangles have equal corresponding angles and proportional corresponding sides.
- AA is enough: two equal angles force the third pair to be equal.
- SAS requires the included angle between the two proportional sides.
- Basic Proportionality Theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio.
- If the side ratio is k, the perimeter ratio is k and the area ratio is k².
- Corresponding medians and altitudes are also in the ratio k.
Worked example
Triangles with sides 3, 4, 5 and 6, 8, 10 are similar by SSS with scale factor k=2. Their perimeters are 12 and 24, ratio 2. Their areas are 6 and 24, ratio 4=2². This confirms that area ratio is the square of the side ratio, not the side ratio itself.
Traps
Write vertices in corresponding order; otherwise the proportions are wrong. SSA is not a similarity condition. In SAS, the equal angle must be between the proportional sides. The area ratio is k², not k. BPT applies when the line is parallel to one side, and the segments are on the two other sides. Do not assume triangles are similar from a diagram alone.