Question

Shown below is an 18 by 15 rectangle with an arrangement of smaller squares within it. Each of the smaller squares has a side length of ‘d’ units. Find d.

Thank You.

Answer 1

Given :- Shown below is an 18 by 15 rectangle with an arrangement of smaller squares within it. Each of the smaller squares has a side length of ‘d’ units. Find d.

Solution :-

from rough diagram we get, since all squares are with sides as d units .

In ∆DEA ,

→ sin θ = EA / DA

→ sin θ = EA / d

→ EA = d•sin θ

and, in ∆ABC ,

→ cos θ = AB / AC

→ cos θ = AB / 3d

→ AB = 3d•cos θ

then,

→ EA + AB = EB

→ EB = FI

→ d•sin θ + 3d•cos θ = 15 ----------- Eqn.(1)

similarly, we get,

→ 2d• cos θ + d•sin θ + d•cos θ + d•sin θ = 18 ------- Eqn.(2)

subtracting Eqn.(1) from Eqn.(2),

→ 3d•cos θ - 3d•cos θ + 2d•sin θ - d•sin θ = 18 - 15

→ d•sin θ = 3 ------ Eqn.(3)

putting Eqn.(3) in Eqn.(1) ,

→ 3 + 3d•cos θ = 15

→ 3d•cos θ = 15 - 3

→ d•cos θ = 12/3

→ d•cos θ = 4 ----------- Eqn.(4)

squaring Eqn.(3) and Eqn.(4) now, and adding them we get,

→ (d•sin θ)² + (d•cos θ)² = (3)² + (4)²

→ d²•sin²θ + d²•cos²θ = 9 + 16

→ d²(sin²θ + cos²θ) = 25

→ d² = 25

→ d = 5 units (Ans.)

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