Question

The figure below shows a triangular wooden frame ABC. The side AD of the frame has rotted and needs to be replaced:

Answer 1

Answer:

The question has missing figure, the figure is in the attachment.

The length of wood needed to replace AD is $5.01\ in$.

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 = $30\°+15\°=45\°$

tan∠ACB =$\frac{AB}{BC}$

$tan45\°=\frac{AB}{BC}\\ 1=\frac{AB}{12} \\AB=12\ in$

Now in ΔDCB,

∠DCB = 30° and  BC=12 in

tan∠DCB =$\frac{DB}{BC}$

$tan30\°=\frac{DB}{12}\\\frac{1}{\sqrt{3} } =\frac{DB}{12} \\DB=\frac{12}{\sqrt{3} } =\frac{12\times\sqrt{3} }{3}=4\sqrt{3}$

$DB = 4\times1.732=6.928\ in$

Length of AD = Length of AB - Length of DB

Length of AD= $12-6.928=5.072\approx5.01\ in$

Thus  the length of wood needed to replace AD is $5.01\ in$.