Purely Geometrical Average of Two Numbers

Draw 2 parallel lines, the lengths of which correspond to 2 numbers A and B whose average you wish to find. Connect the endpoints of these lines so as to form a trapezoid/trapezium.

Bisect each of the legs of this trapezoid/trapezium and connect their midpoints. (This line is called the median)

The length of this line is the average of the two numbers A and B.

Though relatively simple, I thought it was so cool how a mathematical concept like average could be represented completely geometrically, without any knowledge of math!

Anybody want to take a stab at proving that the median is indeed the average of the 2 lines?

#Geometry

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Visually if you just cut by the red line the trspezium take upper region rotate 180 degrees and glue the two part putting base B with base A. You will obtain a parallelogram one side measure A+B and the other 2m therefore m=(A+B)/2. Old people figured out this way

Mara Jares - 3 years, 5 months ago

Suppose $ABDC$ is a trapezium, so that the vectors $a = \overrightarrow{AB}$ and $b = \overrightarrow{CD}$ are parallel. So, let $a = \lambda v$ and $b = \mu v$ , for some vector $v$ . Also suppose that $c = \overrightarrow{AC}$ and $d = \overrightarrow{DB}$ . Let $M,N$ be the respective midpoints of $AC$ and $BD$ ; this means that $\overrightarrow{AM} = \overrightarrow{MC} = \frac{1}{2} c$ and $\overrightarrow{NB} = \overrightarrow{DN} = \frac{1}{2} d$ .

So, we have that

$\overrightarrow{MN} = \frac{1}{2} (c + d) + \mu v = - \frac{1}{2} (c + d) + \lambda v .$

Hence,

$2 \overrightarrow{MN} = (\lambda + \mu) v$

and as a result

$\overrightarrow{MN} = \frac{\lambda + \mu}{2} v.$

We understand the quantity $\frac{\lambda + \mu}{2}$ to be the arithmetical mean of $\lambda$ and $\mu$ , so we have to understand that the line segment connecting the midpoints of the non-parallel sides of a trapezium is half the "sum" of the line segments of the parallel sides of a trapezium; note the lack of use of "lengths" in not only the proof, but also the result obtained.

P.S. The origins of our understanding of mathematics are purely geometric (aside from the idea of counting objects); the Greeks had the right idea in thinking geometrically, and it has carried on until the turn of the last century, when mathematics started to go deeply south.

A Former Brilliant Member - 3 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$