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If altitudes of a triangle are in A.P. then sides of the triangle are in

If altitudes of a triangle are in A.P. then sides of the triangle are in Correct Answer H.P.

Concept:

A sequence of numbers is called a harmonic progression if the reciprocal of the terms are in AP. In simple terms, a,b,c,d,e,f are in A.P. if 1/a, 1/b, 1/c, 1/d, 1/e, 1/f are in H.P.

Formula used:

Area of triangle = ½ base × altitude

Explanation:

Let x, y, z be the altitudes to sides a, b, c respectively of a triangle.

We know that, Area of triangle = ½ base × altitude

⇒ ax/2 = by/2 = cz/2

Let ax = by = cz = k

⇒ x = k/a, y = k/b, z = k/c      ----(1)

Since x, y, z are in AP.

⇒ 2y = x + z

On substituting the values from equation (1), we get,

⇒ 2k/b = k/a + k/c

⇒ 2/b = 1/a + 1/c

⇒ a, b, c are in HP

Hence, if a triangle's altitudes are in A.P., then the sides of the triangle are in H.P.

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