Grades 9-10 Video Solutions 2023-2023_9-10

Video Transcription

Points M and N are the midpoints of two sides of the rectangle. What fraction of the area of the rectangle is shaded?  Let's think about this image. So when we look at this image, notice that the area that this region 1 occupies is exactly  1 quarter of this total area, because they are similar triangles, and region 1's height is half of this height, and because they're similar triangles, this length must be half  of this length. Therefore, region 1 is 1 quarter of the total shaded area that it occupies. We can do the same thing with 2, 3, 4, and 5, although they might not be as obvious.  And then, because each of 1, 2, 3, 4, and 5 is a quarter of the region it occupies, and then if we sum up each of these regions, we find that that's just the total rectangle  area, and therefore, the regions 1, 2, 3, 4, and 5 occupy 1 quarter of the total rectangle area, and the correct answer is C, 1 fourth.

Video Summary

The problem involves determining the fraction of a rectangle's area that is shaded. Given that points M and N are midpoints of two sides of the rectangle, the analysis involves identifying smaller regions (1 to 5) within the rectangle. Each of these regions is found to occupy one-quarter of the area they cover due to symmetry and similarity arguments. Summing these areas results in the shaded regions collectively occupying one-quarter of the total rectangle's area. The correct answer to the problem is therefore that the shaded area is \(\frac{1}{4}\) of the rectangle's total area.

Keywords

rectangle

shaded area

midpoints

symmetry

fraction