Question

ABCD is a rectangle. P is the midpoint of BC, AP = 8 cm. Find the sides AB and BC.

Answer 1

ABCD is a rectangle.

P is the midpoint of BC.

∠APB = 60°

AP = 8cm 

AB and BC

ΔABP,

Since ABCD is a rectangle, ∠ABP = 90°

Therefore, ΔABP is a right-angled triangle.

P lies on BC, and is its midpoint, so triangle ABP can be solved using trigonometry.

Using Trigonometry:

Sine Rule in △ABP

sin(∠APB) = `(AB)/(AP)`

sin(60°) `(AB)/8`

 AB = 8 × sin(60°)

AB = 8 `xx sqrt3/2`

AB = `4sqrt3` cm

Find BP using trigonometry or Pythagorean theorem

Find BP using the cosine function:

cos(∠APB) = `(BP)/(AP)`

cos(60°) = `(BP)/8`

BP = 8 × cos(60°)

BP = `8 xx 1/2`

BP = 4 cm

Find BC using the midpoint information

Since P is the midpoint of BC, we have:

BC = 2 × BP

BC = 2 × 4

= 8 cm