Let M be the intersection of the diagonals. Fold the paper along AC and look at the relation between â–ł ADC and â–ł ABC. Does it give you a relationship between BM and DM? Do the same for the other diagonal.
See solution.
Practice makes perfect
Let's begin by drawing a line and marking two points A and B on it.
Next, we will draw a point D that is not on the line, and then let's draw the ray AD.
Now we place the compass tip on A and draw an arc that intersects AB and AD. We will mark the intersection points as X and Y.
With the same compass setting, we put the compass tip on D and draw an arc that intersects AD. Let Z be the intersection point. After that, keeping the same compass setting, we place the compass tip on B and draw an arc that intersects AB.
With the compass, we measure the distance between X and Y. Then, we place the compass tip on Z and draw an arc that intersects the previous arc. With the same compass setting, we repeat this process but with the compass tip on P.
Finally, we draw the line passing through D and W, and we also draw the line passing through B and Q. Let C be the intersection between these two lines.
We have finished the construction of our parallelogram ABCD. Next, we will draw its two diagonals AC and BD.
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
The first thing we need to do is fold the paper along diagonal AC. After this, we will see that △ ADC has been mapped onto △ ABC. This latter fact implies that BM ≅ DM.
M is the midpoint of BD.
Similarly, when we fold the paper along the diagonal BD, we get that △ ABD is mapped onto △ BCD which implies that AM≅ CM.
