a 34 foot tree casts a 53 foot shadow what is the distance from the top of the tree to the far shadow

When the two side lengths of a right triangle are given and we need to know the side length of the other side, we can use the Pythagorean Theorem to compute it!

[tex]a^2+b^2=c^2[/tex],

where a and b are the legs of the triangle and c is the hypotenuse.

This equation can be rearranged to find each side length:

[tex]c=\sqrt{a^2+b^2}[/tex]
[tex]b=\sqrt{c^2-a^2}[/tex]
[tex]a=\sqrt{c^2-b^2}[/tex]

[tex]\hrulefill[/tex]

Solving the Problem

Visualizing the Problem

We're told

a tree has a height of 34 ft
casts a shadow on the ground--adjacent to the tree--of length 53 ft

and we need to find the distance from the tip of the shadow to the top of the tree.

An image of a right triangle can be drawn to model this:

the vertical leg representing the tree's height or 34ft
the horizontal leg representing the length of the shadow or 53 ft
the hypotenuse representing the distance we need to find.

(See the image attached).

[tex]\dotfill[/tex]

Solving for the Distance

We're given the a and b parts of the theorem's equation, all we do is plug and calculate for the hypotenuse or the distance.

[tex]c=\sqrt{34^2+53^2}[/tex]

[tex]c=\sqrt{3965}[/tex]

[tex]c=62.97[/tex]

So, the distance between the tip of the shadow and the top of the tree is 62.97ft.