Triangles
ABCand
CDEare equilateral triangles of the same size. If
AC=10and
∠BCD=80, find the area of triangle
ABF.
Since
∠ACB=60∘,
∠ACD=60∘+80∘=140∘.
Since
△ACDis isosceles,
∠CAD=∠ADC=2180∘−140∘=20∘.
It then follows that
∠BAF=∠BAC−∠DAC=60∘−20∘=40∘.
By theexterior angle theorem,
∠BFA=20∘+60∘=80∘.
By applyingsine lawon
△BAF, we have
sin40∘BF=sin80∘10⟹BF=sin80∘sin40∘(10).
Note that the area of a triangle is half the product of two adjacent sides multiplied by the sine of the included angle. We have
A=21(AB)(BF)(sin60∘)=21(10)(sin80∘sin40∘)(10)(sin60∘)≈28.263.□