Pythagoras theorem word problem A tree is 8 m north and 6 m east of another tree. One of the trees is 12 m tall, and the other tree is 17 m tall. Find the distance between the tops of the trees
You are asked to find the Euclidean distance between two points which are $8\,m$ apart on one axis, $6\,m$ apart on a second axis and $17-12\, m$ on a third. The axis are pairwise orthogonal.
Hint:
Your first task is modeling the given information into a geometric model.
The given numbers can be used to come up with Cartesian coordinates.

Here is a visualisation:




What could be $T_1$ and $T_2$?
What could be "TOP1" and "TOP1B", and "TOP2" and "TOP2B"? 


Solution:

Let us model the information from the task:


  A tree is 8 m north and 6 m east of another tree.


So for the bases of the trees we can use relative coordinates
$T_1 = (0,0)$ and $T_2 = (6, 8)$, letting the $x$-axis point to the east and
and the $y$-axis point to the north.


  One of the trees is 12 m tall, and the other tree is 17 m tall. 


This adds height to the picture. We abstract the trees as line segments from their base at $(x_i, y_i, 0)$ to their their top $(x_i, y_i, z_i)$.
So we have
$$
(0,0,z_1) \quad (6, 8, z_2)
$$
for the two top points.

The text gives the information about the heights, but leaves out which one has which height. So we have either
$$
\text{TOP}_1 = (0,0,12) \quad 
\text{TOP}_2 = (6, 8, 17)
$$
or
$$
\text{TOP}_{1\text{B}} = (0,0,17) \quad 
\text{TOP}_{2\text{B}} = (6, 8, 12)
$$

So lets us look at what to do:


  Find the distance between the tops of the trees


For the first scenario this would be
$$
d = \lVert(0,0,12) - (6, 8, 17)\rVert
$$
for the second scenario
$$
d = \lVert(0,0,17) - (6, 8, 12)\rVert
$$
where the Euclidean norm of a vector
$$
\lVert x \rVert 
= \lVert (x_1, x_2, x_3) \rVert 
= \sqrt{\sum_{i=1}^3 x_i^2}
$$
was used.
So or the first we get
$$
d^2 = (0-6)^2 + (0-8)^2 + (17-12)^2
$$
for the second
$$
d^2 = (0-6)^2 + (0-8)^2 + (12-17)^2
$$
so indeed we do not need the information which tree has what height height, both. In both scenarios we get
$$
d^2 = 36 + 64 + 25 = 125 \Rightarrow d = \sqrt{125} = 11.1803398\dotsb
$$