Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

A square lies in a coordinate system as shown. Each point x comma y on the square is moved to 1 over x comma 1 over y. What will the resulting figure look like?  Here's the figure drawn a little bit bigger. So first let's look at what happens to this line right here. This line has x coordinate equal to 1, so after the transformation it's  still going to have x coordinate equal to 1. Here the y coordinate ranges from 1 to 2, so after the transformation to 1 over y, the y coordinate is going to range from 1  to 1 half. So we're going to get a line segment looking like this. By exactly the same reasoning  we can do this horizontal line segment and it's going to be turned into this blue segment over here. Now we just need to figure out what happens  here. Well, on this top segment the y coordinate is 2, so after the transformation the new  y coordinate has to be 1 half, so it has to be along here. And then the x coordinate starts at 1 and goes to 2, so it ranges from 1 to 1 half. So we get this blue segment right  here and by the similar logic we can figure out what happens to this segment and we get this blue segment here. So our answer is C.

Video Summary

In this transformation, each point \( (x, y) \) on a square is moved to \( \left( \frac{1}{x}, \frac{1}{y} <br />ight) \). The vertical line at \( x = 1 \) remains unchanged horizontally but shifts its \( y \)-values from 1 to 2 into 1 to 0.5. Similarly, the horizontal line at \( y = 1 \) is transformed vertically by shifting \( x \)-values. For the segment where \( y = 2 \), it transforms to \( y = 0.5 \), affecting \( x \)-values from 1 to 0.5, and vice versa for \( x = 2 \). Thus, the transformed shape is a new, smaller square cornered differently, resulting in answer C.

Keywords

transformation

square

coordinates

inverse

geometry