The figure below shows a triangular wooden frame ABC. The side AD of the frame has rotted and needs to be replaced:What is the length of the wood that is needed to replace AD?
Accepted Solution
A:
Answer:The question has missing figure, the figure is in the attachment.The length of wood needed to replace AD is [tex]5.01\ in[/tex].Step-by-step explanation:Given;ABC is a triangle in which AB is of 12 inches in length.∠DCB = 30° ∠ACD = 15°Solution,In ΔABC, BC=12 in∠ACB = ∠DCB + ∠ACD = [tex]30\°+15\°=45\°[/tex]tan∠ACB =[tex]\frac{AB}{BC}[/tex][tex]tan45\°=\frac{AB}{BC}\\ 1=\frac{AB}{12} \\AB=12\ in[/tex]Now in ΔDCB,∠DCB = 30° and BC=12 intan∠DCB =[tex]\frac{DB}{BC}[/tex][tex]tan30\°=\frac{DB}{12}\\\frac{1}{\sqrt{3} } =\frac{DB}{12} \\DB=\frac{12}{\sqrt{3} } =\frac{12\times\sqrt{3} }{3}=4\sqrt{3}[/tex][tex]DB = 4\times1.732=6.928\ in[/tex]Length of AD = Length of AB - Length of DB Length of AD= [tex]12-6.928=5.072\approx5.01\ in[/tex]Thus the length of wood needed to replace AD is [tex]5.01\ in[/tex].