For an equilateral triangle of side $a$ and altitude $h$ , which relation is correct?

Class 10Triangles (Similarity)Geometry — Similarity

A $4 a^{2} = 3 h^{2}$
B $3 a^{2} = 4 h^{2}$
C $a^{2} = 2 h^{2}$
D $2 a^{2} = 3 h^{2}$

Step-by-Step Solution

The altitude of an equilateral triangle bisects the base, so by Pythagoras

h^{2} = a^{2} - (\frac{a}{2})^{2} = \frac{3 a ^{2}}{4}

. Multiplying both sides by 4 gives

4 h^{2} = 3 a^{2}

. "

4 a^{2} = 3 h^{2}

" is the same relation with a and

h

swapped — but

h

is always the SMALLER of the two, so the bigger multiplier must sit with

h

Answer: B)

3 a^{2} = 4 h^{2}

Key idea: Three similarity criteria

**AA (Angle-Angle)**: Two angles equal → similar. (Third angle is automatic by 180° angle sum.)

**SSS (Side-Side-Side proportionality)**: All three pairs of corresponding sides in same ratio → similar.

**SAS (Side-Angle-Side)**: Two pairs of sides proportional AND included angles equal → similar.

**Important**: SSA does NOT prove similarity (just like for congruence).

AA is by far the most-used in problems: spot two equal angles (often via parallel lines or shared angle) and declare similarity.

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For an equilateral triangle of side a and altitude h, which relation is correct?

Step-by-Step Solution

Key idea: Three similarity criteria

Master this chapter in 30 minutes a day.

More Class 10 Triangles (Similarity) practice

For an equilateral triangle of side $a$ and altitude $h$ , which relation is correct?