When working with the different types of rigid motions, it was used that they map a segment into another segment. To verify that this is true, consider a segment AB and any rigid motion. Let C be the image of A and D the image of B.
Because rigid motions preserve distances, CD is equal to AB. Now, to check that every point of AB was actually mapped onto CD, consider a point P on AB different from the endpoints. Let Q be the image of P under the rigid motion.
(I): Segment Addition Postulate
Rigid motions map n-sided polygons onto n-sided polygons.
Consider a rigid motion and a circle
Since rigid motions preserve distances, the image of is a circle whose radius is equal to the radius of Therefore, is the circle centered at and radius r.
The image of is a circle centered at with radius PQ. Since PQ and are equal, is the circle centered at passing through point
Emily and her family are spending the weekend at a lake house. On Saturday afternoon, Emily begins to work on her geometry homework that asks her to determine if the following pair of polygons are congruent.
When two figures are not congruent, no sequence of rigid motions maps one figure onto the other. Even so, it is possible that a rigid motion maps certain parts of the preimage onto their corresponding parts, but not all. For example, consider the following pair of quadrilaterals.
Furthermore, consider the following sequences of transformations.
To determine if any of the given sequences maps ABCD onto PQRS, apply them to ABCD, one at a time, and compare each resulting image to PQRS.
Because of copyright, in marketing, two different companies cannot have the same name or logo. Here, knowing how to determine if two figures are congruent is useful in dealing with intellectual property. Sometimes lesser-known brands use logos almost congruent to those of well-known brands. The intention is to receive instant recognition. For instance, consider the well-known logo of MathLeaks and the logo of a startup company called MathLovers.
Congruent figures are everywhere in daily life. For example, the figures formed by lines on one half of a basketball court are congruent to the figures formed on the other half. However, verifying this congruence through rigid motions is a difficult thing to do, given the circumstance of not being able to see the rigid motions occur.polygons, which will be studied in the next lesson.