Triangles • Topic 2 of 4

Similarity & Area Ratios

Similar triangles have the same shape but possibly different size: corresponding angles are equal and corresponding sides are in a fixed ratio k. The fastest test in CAT is AA — if two angles match, the third must too, so the triangles are similar. The result the exam loves is the area rule: if the sides are in ratio k (say 3 : 5), the areas are in ratio k² (9 : 25). Perimeters, medians, altitudes, inradii and circumradii all scale linearly with k, but areas scale with k². A second high-yield tool is the Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle cuts the other two sides in the same ratio, creating a smaller similar triangle. Whenever you see parallel lines, a midpoint, or two triangles sharing an angle inside a figure, reach for similarity first — it converts an awkward length into a clean proportion in one line.

✅ Solved examples

1. Two similar triangles have corresponding sides in the ratio 3 : 4. Find the ratio of their areas.
Area ratio = (side ratio)² = 3² : 4² = 9 : 16.
2. In triangle ABC, DE ∥ BC with D on AB and E on AC. If AD = 4, DB = 6, AE = 5, find EC.
By BPT, AD/DB = AE/EC ⇒ 4/6 = 5/EC ⇒ EC = 5 × 6/4 = 7.5.
3. The areas of two similar triangles are 50 cm² and 98 cm². If a side of the smaller is 10 cm, find the corresponding side of the larger.
Side ratio = √(98/50) = √(49/25) = 7/5. Corresponding side = 10 × 7/5 = 14 cm.
4. A vertical pole 6 m tall casts a 4 m shadow. A nearby tower casts a 30 m shadow at the same time. Find the tower’s height.
Sun’s rays give similar right triangles, so height/shadow is constant: 6/4 = h/30 ⇒ h = 30 × 6/4 = 45 m.

✏️ Practice — try these, take hints as needed

1. Similar triangles have sides in ratio 2 : 5. Ratio of areas?
Square the side ratio.
2² : 5².
4 : 25.
4 : 25
2. Areas of two similar triangles are 16 : 49. Ratio of corresponding sides?
Sides scale as the square root of areas.
√16 : √49.
4 : 7.
4 : 7
3. DE ∥ BC; AD = 3, DB = 5, AE = 6. Find EC.
Apply BPT.
3/5 = 6/EC.
EC = 6 × 5/3.
10
4. A smaller similar triangle has half the perimeter of a larger one. What fraction is its area?
Perimeter ratio = side ratio = 1/2.
Area ratio = (1/2)².
= 1/4.
One-quarter (1/4)
5. Two similar triangles have areas 45 and 80. The smaller has a median of 9. Find the larger triangle’s corresponding median.
Side/median ratio = √(80/45).
√(16/9) = 4/3.
9 × 4/3.
12

📝 Topic test — 8 questions

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