A ladder manufacturing company manufactures a foldabel step ladder as shown below :

The lengths of legs AB and AC are 120 cm each.

(i) If vertical ∠BAC = 36°, what is the ratio of ∠BAC and ∠ACB ?

(ii) (a) If vertical ∠BAC = 60°, what is the length of BC?

(b) Which type of triangle is △ABC ?

Given,

AB = AC = 120 cm

So, it is an isosceles triangle.

∴ ∠ABC = ∠ACB (Angles opposite to equal sides are equal in a triangle)

Let ∠ABC = x

(i) Sum of angles in a triangle is 180°.

∠BAC + ∠ABC + ∠ACB = 180°

36° + x + x = 180°

2x = 180° - 36°

2x = 144°

x = $\dfrac{144\degree}{2}$

x = 72°

So, ∠ABC = ∠ACB = 72°

∠BAC : ∠ACB = 36° : 72° = 1 : 2.

Hence, ratio = 1 : 2.

(ii) (a) ∠BAC = 60°

Since AB = AC,

So, ∠ABC = ∠ACB

Sum of angles = 180°

∠BAC + ∠ABC + ∠ACB = 180°

60° + 2(∠ACB) = 180°

2(∠ACB) = 180° - 60°

2(∠ACB) = 120°

∠ACB = $\dfrac{120\degree}{2}$

∠ACB = 60°.

∴ ∠ACB = ∠ABC = 60°

Since all the three angles are 60°, so all the three sides must be equal.

∴ BC = AB = AC = 120 cm.

Hence, BC = 120 cm.

(b) Since all angles are 60° and all sides are equal (= 120 cm)

∴ △ ABC is an equilateral triangle.

Hence, △ABC is an equilateral triangle.