What is AB : BC : CA equal to?

Given:

Perimeter of triangle ABC = 105 cm

Ratio of altitudes AD : BE : CF = 3 : 5 : 6

Formula used:

The area of a triangle (Area) can be expressed as: Area = \((\dfrac{1}{2}) × base × height.\)

For a triangle with sides a, b, c and corresponding altitudes ha, hb, hc, we have:

2 × Area = a × ha = b × hb = c × hc

This implies that the sides of a triangle are inversely proportional to their corresponding altitudes. That is, \(a : b : c = (\dfrac{1}{h_a}) : (\dfrac{1}{h_b}) : (\dfrac{1}{h_c})\)

Calculations:

Let the sides of the triangle be a, b, c, where:

a = BC (side opposite to vertex A)

b = CA (side opposite to vertex B)

c = AB (side opposite to vertex C)

Let the altitudes be ha, hb, hc, where:

ha = AD

hb = BE

hc = CF

Given the ratio of altitudes: ha : hb : hc = 3 : 5 : 6.

Using the property that sides are inversely proportional to altitudes:

\(a : b : c = (\dfrac{1}{3}) : (\dfrac{1}{5}) : (\dfrac{1}{6})\)

⇒ \(a : b : c = (\dfrac{1}{3}) × 30 : (\dfrac{1}{5}) × 30 : (\dfrac{1}{6}) × 30\)

⇒ a : b : c = 10 : 6 : 5

This means the ratio of the sides BC : CA : AB is 10 : 6 : 5.

Therefore, AB : BC : CA = c : a : b = 5 : 10 : 6.

∴ The correct answer is option 2.