MHB Find the distance between points C and D if the height of the tree is 4m

[Elissa89]

So the question is...From point C on the ground level, the angle of elevation to the top of a tree is 30 degrees. From point D, which is closer to the tree, the angle of elevation is measured to be 45 degrees. Find the distance between points C and D if the height of the tree is 4m.

I know triangle 1 has angles 30 degrees, 60 and 90. So the adjacent side is 4*sqrt(3)

I know triangle 2 has angles 45 degrees, 45 and 90. The adjacent side is 2*sqrt(2)

From here I am stuck as I do not know how to find the distance between point C and D

 

[MarkFL]

I would let \(d_C\) be the distance from point C to the tree in meters, and so we may write:

$$\tan\left(30^{\circ}\right)=\frac{4}{d_C}\implies d_C=4\cot\left(30^{\circ}\right)=4\sqrt{3}\quad\checkmark$$

Likewise for point D:

$$\tan\left(45^{\circ}\right)=\frac{4}{d_D}\implies d_D=4\cot\left(45^{\circ}\right)=4$$

We should expect that in a 45-45-90 triangle the adjacent and opposite sides are equal. And so the distance \(d\) between the two points is:

$$d=d_C-d_D=?$$