How to do these problems?

Question

To determine whether you can see a bird in the sky, you are trying to compare it to the height of a tree

nearby. You place a small mirror on the ground so you can see the reflection of the top of the tree. Your

eyes are 5 feet above the ground. The distance from you to the mirror is 12 feet and the distance from

the mirror to the tree is 39 feet.

If the bird is flying behind the tree at 16 feet, is the bird flying higher or lower than the top of the tree?

Answer 1

Draw the diagram as described by Sindhuja R.

The angle of refraction is equal to the angle of incidence..

The two right triangles are similar.

t represents the height of the tree

t : 5 :: 39 : 12

12t = (5)(39)

12t = 195

t = 16.25

Answer 2

First you need to draw a figure, i'll explain in simple steps for you to draw.

Firstly, trying to understand the question:-

There is a tree, and a bird is behind the tree, assuming tree is towards right of you. We place a mirror in between tree and us, so that through reflection, we can see the top of the tree.

Draw: They have given, eyes are 5 feet above the ground, draw a vertical line, label it 5 feet. let's name it 'EF', F on the ground and E on the top.

The distance from you to mirror is 12 feet, draw a horizontal line to a point, let's name it G. So, FG=12.

Join EG, so that we get a triangle EFG.

And the distance from the mirror to the tree is 39 feet. draw horizontal line to a point, let's say C. So, GC=39.

So, as we place a mirror, we can see the top of the tree in the mirror. For this to understand conceptually, assume there is a lake near the tree, you could see the reflection of the tree in the lake, below the ground level. In the same way, when we place a mirror we can see the image below the ground level.

So, to draw that, let's say D be a point on the top of tree. draw this below the ground level.

And join GD and DC. we get a triangle GDC.

As GDC is the image in the mirror, it is same as original we can draw same image above the ground.

Take a point A equidistant from D above the ground level, we get triangle GAC.

Mark a point B on AC so that EF=BC= 5 feet. and AB= 'x' feet(assume).

Now, coming to the solution:-

Angle EGF = angle DGC (As, they are vertically opposite angles).

So, angle DGC= angle CGA (As, this is image, both will be same). mark all of the with 'a'.

Now, in triangle EFG=> taking tan a = EF/ FG {Opposite/Adjacent}

tan a = 5/12

a = tan ^-1 (5/12) ------------1

angle CGA = a = tan ^-1 (5/12).

Now in triangle GAC => tan a = AC/GC

tan a = (x+5) / 39

a = tan ^-1 ( (x+5) /39 ) ---------- 2

from equations 1 and 2

tan ^-1 (5/12) = tan ^-1 ( (x+5) /39 )

cancelling tan ^-1 on both the sides.

5/12 = (x+5) /39

Calculating x, we get => x = 11.25

so, AC= x+5 => 11.25+ 5 = 16.25 feet.

Bird is flying at 16 feet height, but the tree is 16.25 feet height.

Hence, bird is flying below the top of the tree.

Hope this help! for any further doubts, i am available to clear any sort of doubts.