Grades 7-8 Video Solutions 2021-video 2021 7-8/30

Video Transcription

Question 30. The diagram shows a quadrilateral divided into four smaller quadrilaterals with a common vertex K. The other labeled points divide the  sides of the large quadrilateral into three equal parts. The numbers indicate the areas of the corresponding small quadrilaterals. What is the area of the  shaded quadrilateral? If we take a look at our quadrilateral we can start to draw additional lines to break it up further. We draw a line here and here. Now  we have to remember one important thing that the question tells us. That each of the sides of the large quadrilateral is broken up into three even parts. Now if  we also remember that the area of a triangle is base times height divided by two then we can begin to solve this problem. Let's call this area A and then  since we know that the base of the triangle next to it is twice as long as the base of the other triangle we can label this as 2A. Next we can draw a line  over here and do the same thing. Call this triangles area B and the one next to it 2B since the base of that triangle is twice the length of the base of the  other triangle and their heights are the same. We can keep doing this. We have C and 2C and finally 2D and D. With this we are set up to start making some  formulas. First we take a look at this quadrilateral 2B plus 2A. We know in the problem it is given that the area of this quadrilateral is 18 so we know that  2A plus 2B is equal to 18. We can factor out a 2 and divide both sides by 2 and we get A plus B equal to 9. Next we have this quadrilateral B plus C. From the  given problem we know that this quadrilateral is 8 so B plus C equals 8. Finally we have this quadrilateral the last given one 2C plus 2D and it is  given that its area is 10 so 2C plus 2D is equal to 10. Again we can factor out a 2 and divide both sides and we get C plus D equals 5. From this we know that A  plus B plus C plus D equals 9 plus 5 or 14. Now since we are trying to find the final quadrilateral we can replace B and C for 8 as we already know this to  be true so A plus 8 plus D equals 14. We subtract 8 from both sides and get A plus D equal to 6. We notice that the last part that we haven't counted yet  the shaded quadrilateral has area of A plus D so with this we get our answer C 6

Video Summary

The problem involves a large quadrilateral divided into smaller quadrilaterals with a common vertex K, where sides are divided into three equal parts. By analyzing given areas and relationships between triangles, we derive key equations: \(2A + 2B = 18\), \(B + C = 8\), and \(2C + 2D = 10\). Solving these, we find \(A + B = 9\), \(C + D = 5\). Seeking the shaded quadrilateral's area, where \(A + D = 6\), we conclude the shaded area is 6. Hence, the answer is 6.

Keywords

quadrilateral

area calculation

vertex K

equations

shaded area