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

Video Transcription

A large square is divided into two unequal squares and two equal rectangles as shown. The vertices of the shaded quadrilateral are the midpoints of the sides of the two squares.  The area of the shaded quadrilateral is 3. What's the area of the unshaded part of the largest square?  So let's just look at this one triangle here. We can find seven more copies of this triangle in this square right here.  So the area of this square is 8 times the area of this little triangle. Well we can do the same thing for the other three pieces of the shaded region as well,  and that tells us that the area of the entire square is 8 times the area of the shaded region.  So since the shaded region has area 3, the entire square has area 24, and so the unshaded portion has area 21. There's actually another way to do this problem as well.  If we go back to this diagram, it turns out it doesn't actually depend where these two lines are drawn, so we can just draw them to make everything equal, and using this we  can just calculate the side length of the square to be the square root of 24, and then go from there.

Video Summary

The problem involves a large square divided into two unequal squares and two equal rectangles, with a shaded quadrilateral whose vertices are midpoints of the square sides. The given area of the shaded quadrilateral is 3. It is determined that this shaded area represents 1/8 of the total square's area, thus the entire square has an area of 24. Consequently, the unshaded portion of the square is 21. The solution also notes that the problem can be approached by considering the square's side length derived from the total area of 24.

Keywords

geometry

shaded quadrilateral

square area

midpoints

unequal squares