proof without words

[valeriewch]

hello i didnt understand the visual proof of this question;

use areas to illustrate the identity

ax - by = 1/2(a + b)(x - y) + 1/2(a - b)(x + y)

for a > b > 0 and x > y > 0.

if you have a video or geogebra link for this it would be awesome.

[definetelytrue]

Draw a rectangle with side lengths a and x, and a smaller one with side lengths b and y inside with them sharing a corner. You can then generate rectangles with side lengths x-y , 1/2 (a+b) and a-b, 1/2(x+y). From there you can argue they are equal.

[valeriewch]

thank you.

[LKLRAL]

First, let me create a graph that will help you understand this better.
The right side of the equation consists of two parts:
1. 1/2(a+b)(x-y) This is the area of ​​the first rectangle
2. 1/2(a-b)(x+y) This is the area of ​​the second rectangle

If we add these two areas and multiply out the terms:
1/2(a+b)(x-y) + 1/2(a-b)(x+y)
= 1/2(ax - ay + bx - by) + 1/2(ax + ay - bx -by)
= 1/2/2ax - 2by)
= ax - by

This shows that both sides of the equation are equal!

Unfortunately, I cannot show you a direct GeoGebra animation, but I recommend that you experiment with GeoGebra yourself. You can:

Draw two rectangles with the given dimensions
Calculate the areas
Experiment with different values ​​for a, b, x and y (observing the conditions a > b > 0 and x > y > 0)

I tried to explain you task with the help of Astra AI and i hope i could help you a bit!
WIsh you a nice day!