Question

In the trapezium ABCD, AB || DC and ∠C = 60°, ∠D = 45°. If AB = 6 cm and BC = 4 cm, find

1. The height of the trapezium 
2. Length of DC

Answer 1

Given:

Trapezium ABCD, where AB || DC

∠C = 60°, ∠D = 45°

AB = 6 cm, BC = 4 cm 

Perpendiculars from points A and B to line DC.

AP ⊥ DC

BQ ⊥ DC

So, height of the trapezium = AP = BQ (since AB || DC)

(i) Height of the trapezium:

In ΔBAQ

hypotenuse BC = 4 cm

Angle ∠C = 60° 

Height BQ = h

sin(60°) = `"BQ"/"BC"`

= `sqrt3/2 = h/4`

= `h = 4 ⋅ (sqrt3)/2`

= `2sqrt3` cm

(ii) Length of DC:

In ΔAPD:

∠D = 45°

Height = AP = `2sqrt3`

tan (45°) = `(AP)/(PD) = 1`

PD = AP

= `2sqrt3`

In ΔBQC:

tan(60°) = `"BQ"/"QC" = sqrt3`

BQ = `2sqrt3`

= QC = `(2sqrt3)/sqrt3 = 2`

DC = PD + AB + QC

DC = `2/sqrt3 + 6 + 2`

= 8 + `2sqrt3` cm