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If the 3 altitudes of a triangle are equal, then triangle is:

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A scalene triangle

An isosceles triangle

An equilateral triangle

None of these

answer is C.

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Detailed Solution

Given that the three altitudes of the triangle are equal.
So, AD = BF = CE.
Question Image

Consider triangles BEC and BFC

EC = BF

(Equal altitudes)

∠ BEC = ∠ BFC = 90 o

(BE and BF are Altitudes)

B C = B C

(common)

Δ B E C ≅ Δ B F C

(RHS postulate)

∠ ABC = ∠ BCA

(Corresponding angles)
Also,
Consider the

Δ A D B

and

Δ A D C

∠ ADB = ∠ ADC = 90 o

(AD is altitude)

A D = A D

(common)

∠ ABC = ∠ BCA

(proved above)
Thus, △ADB ⩭ △ADB (ASA postulate)

AB = AC

(Corresponding sides are equal)
Similarly, we can prove △BFC ≅ △BFA.
And, hence,

B C = A C

.
Thus,

A B = A C = B C

.
It implies that if the altitudes of the triangle are equal, then 3 sides of the triangle are also equal. So, the triangle will be an equilateral triangle.
Hence, the correct option is 3.

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