Pythagoras theorem word problem

Question

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

Answer 1

First find out how far apart the trunks of the trees are. Use pythagorean thereom.

Then imagine two monkeys climbing the trees at the same pace. As soon as the monkey on the shorter tree reaches the top he shouts "stop". Then the monkey climbing the taller tree ties a red ribbon around the taller tree. That red ribbon is 12 m off the ground.

Now how far apart is the red ribbon from the top of the shorter tree? (You know how far apart the bases of the trees are. So how far apart are two points on the trees if the two points are both the same distance off the ground?)

Now how far apart is the ribbon on the tall tree from the top of the tall tree? (The ribbon is 12 m off the ground. The tree is 17 meters high.)

Now figure out how far apart the tops of the trees are. Use the pythagorean theorem. You know how far apart the top of the short tree is from the ribbon. And you know how far apart the ribbon is from the top of the tall tree. So how far apart is the top of the share tree to the tall tree?

Answer 2

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 $$ d^2 = 36 + 64 + 25 = 125 \Rightarrow d = \sqrt{125} = 11.1803398\dotsb $$

Answer 3

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.