Similar triangles

Skills and methods in this topic, easy to hard

1. 1Corresponding sides of similar trianglesMatch the letters in order; big side = small side × the scale

   Method△ABC ~ △PQR, PQ ÷ AB = 12 ÷ 4 = 3, QR = 5 × 3 = 15

   Another way: Write a proportionx5 = 124, 4x = 60, x = 15

2. 2Conditions for similar trianglesAAA, 3 sides proportional, 2 sides proportional with the included angle

   Method64 = 96 = 128 = 1.5: 3 sides proportional

   Another way: Find the third angle first180° − 50° − 70° = 60°: both are 50°, 60°, 70°, AAA

3. 3Parallel lines and similar trianglesA shape: AD goes with AB (not DB); hourglass: AB goes with CD

   MethodAD = 2, DB = 3, BC = 10: DE = 10 × 25 = 4

   Another way: Write a proportionDE10 = 25, 5 × DE = 20, DE = 4

4. 4Proving triangles similarOne pair a line, with its reason; common ∠, vert. opp. ∠s, parallels

   Method∠A = ∠A (common ∠), ∠ADE = ∠ABC (corr. ∠s), △ADE ~ △ABC (AAA)

   Another way: By two sides and the angle betweenADAB = 39 = 13, AEAC = 412 = 13, ∠A common

5. 5Similar polygonsAll angles equal, all sides in proportion; angles aren't multiplied

   Methodk = 9 ÷ 6 = 1.5, QR = 4 × 1.5 = 6, ∠Q = ∠B

   Another way: Write a proportionQR4 = 96, 6 × QR = 36, QR = 6

6. 6Areas of similar figuresArea ratio = side ratio²; perimeter ratio = side ratio

   MethodSides 2 : 5, area 8 × 254 = 50

   Another way: By a proportion4 : 25 = 8 : A, A = 8 × 25 ÷ 4 = 50

7. 7The Sierpinski triangleEach step: count × 3, area × 34, side ÷ 2

   MethodStep 3: 3³ = 27 triangles, area (34)³ = 2764

   Another way: Step by step1 → 3 → 9 → 27; 14 × 14 × 14 = 164, 27 × 164 = 2764

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Corresponding sides of similar triangles →

   In the picture, △ABC ~ △XYZ. x = , y =

   Fill in each blank

   ▶Show the answer and working

   Answer: 16, 14

   Scale = XY ÷ AB = 8 ÷ 4 = 2: x = 8 × 2 = 16, y = 7 × 2 = 14.

2. Example 2Proving triangles similar →

   In the picture, ∠ADE = ∠ACB. Which triangle is △ADE similar to, its vertices in matching order?

   • A. △ACB
   • B. △BCA
   • C. △ABC
   • D. △CAB

   Choose the right answer

   ▶Show the answer and working

   Answer: A. △ACB

   ∠A is common and ∠ADE = ∠ACB, so A goes with A, D with C, E with B: △ADE ~ △ACB (AAA).

3. Example 3The Sierpinski triangle →

   The picture shows one step of the Sierpinski triangle. Drawing 2 more steps from it, there are small shaded triangles.

   ▶Show the answer and working

   Answer: 243

   The picture has 27 shaded triangles; each step turns every one into 3: 27 × 3² = 243.