Practice30Create account

In ΔABC right-angled at A, AD is the altitude to the hypotenuse BC. If BD = 5 cm and CD = 20 cm, find ar(ΔABD) : ar(ΔACD).

Class 10Triangles (Similarity)Geometry — Similarity
  • A1 : 4
  • B1 : 2
  • C1 : 16
  • D4 : 1

Step-by-Step Solution

Δ and Δ stand on and with the same height , so their areas are in the ratio 5 : 20 = 1 : 4. The same answer follows from similarity: and , so : 2 and the area ratio is the square, 1 : 4. 1 : 2 quotes the side ratio instead of its square; 1 : 16 squares the ratio a second time; 4 : 1 reverses the order named in the question.
Answer: A) 1 : 4

Key idea: Areas of similar triangles

**Theorem**: The ratio of the areas of two similar triangles is equal to the SQUARE of the ratio of their corresponding sides.

If ΔABC ~ ΔDEF with scale factor , then
Area(ABC)/Area(DEF) = .

**Why**: area scales as length × length. If both length and breadth scale by , area scales by .

**Example**: Two similar triangles have corresponding sides in ratio 3:5. Area ratio = 9:25.

**Reverse**: if areas are in ratio 16:25, sides are in ratio 4:5 (square root).

Master this chapter in 30 minutes a day.

Don't just memorise the answer — build a daily practice habit.

Practice 5 Similar Questions Free →

All Triangles (Similarity) practice questions

More Class 10 Triangles (Similarity) practice

In ΔABC, the bisector of ∠A meets BC at D. If AB = 7.5 cm, BD = 5 cm and DC = 6 cm, find…Two right-angled triangles are given. Which single extra fact is enough to guarantee…In ΔLMN, LM = 20 cm, MN = 21 cm and NL = 29 cm. A student says the right angle is at N,…ΔABC ~ ΔPQR with scale factor k = AB/PQ. Which of these quantities is NOT in the ratio k?A carpenter wants to check whether a cross-bar is parallel to the base of a triangular…