Prove similar

A standard method from Similar triangles: the steps and worked examples.

Steps

Name the triangles in matching order (△ADE ~ △ABC).
Give the equal angles or ratios, one per line, each with a reason.
Finish with the condition: "∴ △ADE ~ △ABC (AAA)".

Worked examples

Example 1: Prove that △PQR and △XYZ are similar, given PQ = 4, QR = 6, RP = 8, XY = 6, YZ = 9 and ZX = 12.

XY/PQ = 6/4  = 3/2
YZ/QR = 9/6  = 3/2
ZX/RP = 12/8 = 3/2
∴ △PQR ~ △XYZ   (3 sides proportional)

Shortest with shortest, longest with longest: that is how the matching order is found.

Example 2: Prove similar, then find a length: AC and BD meet at O, with OA = 4 cm, OC = 6 cm, OB = 6 cm, OD = 9 cm and AB = 5 cm. Find CD.

∠AOB = ∠COD          (vert. opp. ∠s)
OA/OC = 4/6 = 2/3
OB/OD = 6/9 = 2/3
∴ △OAB ~ △OCD        (ratio of 2 sides, inc. ∠)

AB/CD = OA/OC
5/CD  = 2/3
CD    = 7.5 cm

Example 3: Area and perimeter of similar triangles: corresponding sides are 3 cm and 5 cm. The smaller triangle has perimeter 12 cm and area 18 cm². Find the perimeter and area of the larger one.

side ratio = 3 : 5
perimeter  = 12 × 5/3      = 20 cm
area       = 18 × (5/3)²   = 18 × 25/9 = 50 cm²

For the whole topic, read Similar triangles: key points, common mistakes and how the DSE tests it.

Steps

Worked examples

Other methods in this topic