Question

AM, BN, CQ are the altitudes of triangle ABC and are equal. Prove that triangle ABCD is an equilateral triangle.

Answer 1

Answer:

We are given that AM, BN & CQ are altitudes of equal length, of sides BC, AC & AB respectively i.e.,

AM = BN = CQ  

And,

AM ⊥ BC

BN ⊥ AC

CQ ⊥ AB

Step 1:

Let’s consider in ∆ AMC and ∆ BNC,  

∠ C = ∠C [common angles]

∠AMC = ∠BNC = 90° [∵ BN & AM are perpendiculars to AC & BC]

AM = BN [given]

∴ By AAS criterion, we get

∆ AMC ≅ ∆ BNC

∴ AC = BC [∵ Corresponding parts of congruent triangles] ……. (i)

Step 2:

Let’s consider in ∆ AQC and ∆ ABN,

∠A = ∠ A [common angles]

∠ANB = ∠AQC = 90° [∵ BN & CQ are perpendiculars to AC & AB]

BN = CQ [given]

∴ By AAS criterion, we get

∆ AQC ≅ ∆ ABN

∴ AC = AB [Corresponding parts of congruent triangles] ……. (ii)

Step 3:

From (i) & (ii), we get

AB = BC = AC

Since triangles with all its sides having equal length is said to be an equilateral triangle

∴ ∆ ABC is an equilateral triangle

Hence proved