Grades 7-8 Video Solutions 2023-2023_7-8

Video Transcription

Question 28. A regular hexagon is divided into four quadrilaterals and one smaller regular hexagon.  The area of the shaded region and the area of the small hexagon are in the ratio 4 to 3. What is the ratio of the area of the small hexagon to the area of the big hexagon?  Let's draw out our shape. Now, since we know that these are two regular hexagons and then we divide into four quadrilaterals,  we know that the areas of the quadrilaterals will be equal to the one corresponding on the opposite side.  So let's start off by getting the ratio of the shaded region to the small regular hexagon. So we know that this is a ratio of 4 to 3.  So let's put in the small hexagon an area of 3 units. Next, we have the unshaded areas. Since we know that these will be equal to the shaded areas,  this will also be in the ratio of 4 to 3 to the small hexagon. Now, if we assume that both of those areas are 4, and now we make it one entire area,  let's call the area of this shaded region 8. So the ratio of the small hexagon to the big hexagon will be 3 to 8 plus 3. The total of the big hexagon includes the small hexagon.  When we do 3 to 8 plus 3, we get our answer, which will be a 3 to 11.

Video Summary

To solve the problem of finding the ratio of the area of the small hexagon to the big hexagon, a regular hexagon is divided into four quadrilaterals and one smaller regular hexagon. The shaded region's area and the small hexagon share a ratio of 4 to 3. Assuming the small hexagon has an area of 3 units, the corresponding unshaded areas mirror the same ratio. If the shaded regions together have an area of 8, combining all gives a total area for the big hexagon. Therefore, the ratio of the small hexagon to the big hexagon is 3 to 11.

Keywords

hexagon

area ratio

geometry

quadrilaterals

shaded region