Question

In the rectangle ABCD, the diagonals intersect at O. If ∠COD = 120° and AB = 6 cm, OP is perpendicular to AB. Find x, y, BC and AC.

Answer 1

Given:

Rectangle ABCD, with diagonals intersecting at O.

∠COD = 120°

AB = 6 cm → implies AP = PB = 3 cm (since diagonals bisect each other).

OP ⊥ AB.

Required: Find values of x, y, BC, and AC.

Step 1: Understand Triangle AOP

In triangle AOP:

• Right-angled at P
• ∠AOP = 60^°
• AP = 1 cm

Step 2: Use trigonometry to find x = OP

We use: 

tan(60^∘) = `"opposite"/"adjacent" = (OP)/(AP)`

Substitute: 

tan(60^∘) = `sqrt3

`x/1 = sqrt3`

= `x = sqrt3` cm

Step 3: Use trigonometry to find y = AO

We use: 

sin(60∘) = `"opposite"/"hypotenuse" = x/y`

Substitute:

`sqrt3/2 = sqrt3/y`

y = 2

y = `2 sqrt3` cm

Step 4: Find BC

In a rectangle:

Opposite sides are equal

BC = AD

AD = 2x = 2 · `sqrt3 = 2 sqrt3`

Step 5: Find diagonal AC

In a rectangle:

Diagonal are equal

AO = `2sqrt3 = AC = 2 * AO = 4sqrt3` cm