What other length relationships would be interesting to explore? Add your ideas here, or add a discussion by clicking on the DISCUSSION tab at the top of the page.

In order to show that this figure is a square, you must show that (1) this figure is equilateral, and (2) that this figure has four right angles.

On a coordinate grid, you accomplish (1) by using the distance formula or the Pythagorean Theorem. You accomplish (2) by showing the slopes of adjacent sides are negative reciprocals, or that the products of the slopes of adjacent sides are -1.

I know you can look at the figure show at left and it looks as if the sides are the same length, and it looks like the angles are right angles. Nevertheless, lets SHOW this algebraically.

Applying the distance formula

to side AB, we have

If you applied the distance formula to the remaining sides (and YOU SHOULD), you would find the the length of each is equal to the length of side AB.

Next, we look at the slope of adjacent sides AB and BC. The slope of side AB is

Likewise, the slope if BC is

The product of these two slopes is -1, thereby confirming that angle B is a right angle.

Once you have established that the four sides have the same length, and that each angle is a right angle, you have shown (or proved) that the figure is a square.

What other length relationships would be interesting to explore? Add your ideas here, or add a discussion by clicking on the DISCUSSION tab at the top of the page.

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Problem 1, Page 6

Double-click on the applet to open it in a separate GeoGebra window

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Problem 11, Page 8

On a coordinate grid, you accomplish (1) by using the distance formula or the Pythagorean Theorem. You accomplish (2) by showing the slopes of adjacent sides are negative reciprocals, or that the products of the slopes of adjacent sides are -1.

I know you can look at the figure show at left and it looks as if the sides are the same length, and it looks like the angles are right angles. Nevertheless, lets SHOW this algebraically.

Applying the distance formula

to side AB, we have

Next, we look at the slope of adjacent sides AB and BC. The slope of side AB is

Likewise, the slope if BC is

The product of these two slopes is -1, thereby confirming that angle B is a right angle.

Once you have established that the four sides have the same length, and that each angle is a right angle, you have shown (or proved) that the figure is a square.