Question

A carpenter makes stools for electricians with a square top of side 0.5 m and at a height of 1.5 m above the ground. Also, each leg is inclined at an angle of ${60}^{\circ }$ to the ground. Find the length of each leg and also the lengths of two steps to be put at equal distances.

Answer 1

Let AC & BD be the length and AB be the edge of the top of the top of the stool.

AP = 1.5 m

AJ = JK = KP = $\frac{1.5}{3}=0.5m$

(i) ΔAPC

∠P = 90°

sin C = $\frac{AP}{AC}$

⇒ sin60° = $\frac{1.5}{AC}$

_=> $\frac{\sqrt{3}}{2}$ = $\frac{1.5}{AC}$

⇒ AC = $\frac{3}{\sqrt{3}}$

⇒ AC = 1.732 m

In ΔAPC & ΔBQD

∠C = ∠D, ∠P = ∠Q, AP = BQ

ΔAPC ≅ ΔBQD [By RHS congruence criterion]

In ΔAJR

Tan 60° = $\frac{AJ}{RJ}$

⇒ RJ $=\frac{AJ}{\sqrt{3}}$

⇒ RJ = 0.28 m.

(ii) ΔAJR ≅ ΔDBJ'S

RJ = J'S

RS = RJ + JJ' + J'S

= 0.28 + 0.5 + 0.28

= 1.06 m

(iii) In ΔAKL,

tan 60° = $\frac{AK}{LK}$

$\sqrt{3}=\frac{1}{LK}$

LK = $\sqrt{3}$ m

Similarly, ΔAKL ≅ ΔBK'M

LK = K'M

Length of step LM = LK + KK' + K'M

= $\sqrt{3}$ + 0.5 + $\sqrt{3}$

= 3.96 m

Leg AC = leg BD = 1.732 m

Length of step RS = 1.06 m

Length of step LM = 3.96 m