Express the lengths in terms of x

Question

Answer 1

In △ABC

Calculate the angle C

In triangle sum of the angles is 180°

Angle C =180-45-45 =90

Apply sine rule for △ABC

x/sin45°=AB/sin90°=BC/sin45°

x/sin45°=AB/sin90°

x/(1/√2)=AB/1

AB=√2x

x/sin45°=BC/sin45°

BC =x

-------------------

In △BCD

Apply sine rule for △BCD

BC/sin90°=BD/sin30°=CD/sin60°

x/sin90°=BD/sin30°      (Since  BC=x)

x=BD/(1/2)

BD =x/2

x/sin90°=CD/sin60°

x=CD/(√3/2)

CD =(√3/2)x

Answer 2

In a 45°-45°-90° triangle, the length of the hypotenuse is √2 times the length of a leg.

In ∆ ABC angles are 45^o-45^o-90^o, so the lengths ratio is 1 : 1 : √2.

The length of the AC (opposite to the angle B = 45^o) = x.

The length of the AB (opposite to the angle C = 90^o) = √2 x.

The length of the BC (opposite to the angle B = 45^o) = x.

 

In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is √3 times the length of the shorter leg

In ∆ CBD angles are 30^o-60^o-90^o, so the lengths ratio is 1 : √3 : 2.

The length of the BC (opposite to the angle D = 90^o) = x.

The length of the BD (opposite to the angle C = 30^o) = x/2.

The length of the CD (opposite to the angle B = 60^o) = x/√3.

 

CHECK :

Law of sines : BC/sinD = BD/sinC = CD/sinB.

(x)/sin(90^o) = BD/sin(30^o)

BD = x/1 * sin(30)

BD = x/1 * 1/2

BD = x/2