Grades 7-8 Video Solutions 2025-2025_7-8

Video Transcription

Problem number 24. Rhea draws four squares side by side as shown. What is the area of the shaded  quadrilateral? To solve this problem, let's extend the sides of the rectangle to fill out the entire  region and figure out all of the side lengths in the diagram. For example, we know that this side  of the rectangle is 9, which is 3 plus 4 plus 2. Similarly, we can figure out the side length of  this square as 2, because we already know the other side length of this square, and we also  know these two are 6, since this square has side length 4 plus 2 equals 6. We can also fill these  out as 3, from the square of side length 3 and the square of side length 6 that we just discovered.  We also know this is of length 7, since the bottom side of the square has length 3 plus 4 equals 7.  Finally, we could figure out that BG is 5, since we can use 7, which is the length of the larger  square, minus 2, which is the length of the smaller square, and subtract those to get 5,  the length of BG. Now, we can start to figure out some of the areas. We know that the area of the  entire large rectangle that we created, AEFG, has side lengths 9 and 13, so the area is 9 times 13,  or 117. By the way, the 13 is from the length FG, which is 5 plus 2 plus 6 equals 13. Now,  we can start to subtract the corners. Since we know the area of the triangle is 1 half times  base times height, we can use that to figure out the area of the corners. And so, by subtracting  out the corners from the large rectangle, we will get the area of the shaded quadrilateral.  First, let's look at AGB. We know that it has height 5 and base 9, so the area is 5 times 9 over 2,  which is 22.5. Similarly, triangle DEC has height and base 3, so its area is 4.5. And triangle BCF  has height 8 and base 6, so the area is 8 times 6 over 2, or 24. Now, we just subtract out the  corners. 117 minus 22.5 minus 4.5 minus 24 will equal 66. And that is the area that's left over  after we take the large rectangle and remove the corners, which will leave us with just the shaded quadrilateral. And that is the area of 66, so the answer is C, 66.

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