Question

If all the altitudes of the triangle are equal, then the triangle is always a/an

• A. Isosceles triangle
• B. Scalene triangle
• C. Right-angled triangle
• D. Equilateral triangle

Answer 1

The correct option is D Equilateral triangle


Let ΔABC be a triangle with altitudes AD, BE and CF such that AD = BE = CF.
In ΔBEC and ΔCFB,
∠BEC = ∠CFB (Each 90°)
BC = BC (Common)
BE = CF (Given)
∴ ΔBEC ≅ ΔCFB (RHS congruence rule)
⇒ ∠ECB = ∠FBC (CPCT) …..(i)
Now, in ΔADB and ΔADC.
∠ADB = ∠ADC = 90° (Each 90°)
AD = AD (Common)
∠ABC = ∠ACB [From (i)]
∴ ΔADB ≅ ΔADC (AAS congruence rule)
⇒ AB = AC (CPCT) …..(ii)
Similarly, ΔBEC ≅ ΔBEA
⇒ BC = AB (CPCT) …..(iii)
From (ii) and (iii), we get
AB = AC = BC
Therefore, ΔABC is an equilateral triangle.
Hence, the correct answer is option (d).