Grades 9-10 Video Solutions 2022-2022_9-10

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Question 18. The diagram shows square PQRS of side length 1. The midpoint of RS is marked U, and the center of the square is marked W. Line segments TW, UW, and VW split the square  into three regions of equal area. What is the length of SV? Let's take a closer look at our square. We know that the side length is 1. Since we know that RS  has a midpoint of U, we know that this will be half of the side length, so SU will be equal to  1 half. And we know that the center of the square is marked W, so line WU will also be 1 half.  Next, we know that the square is split into three equal regions. We know that the area of the square  is 1, since it has a side length of 1. And if we know that it is 1 third of the area, then we know  that trapezoid SUWV will have an area of 1 third. To figure out the length of SV, we simply have to  take the formula for the area of a trapezoid. We input the numbers that we know, and we get 1 half SU  times the sum of SV and WU equal to 1 third. We can put in the values that we know, knowing SU and WU  are each equal to 1 half. Then we can divide both sides by 1 half times 1 half, getting SV plus 1 half  equal to 4 thirds. Then we can make these denominators the same, and get SV plus 3 sixths equal to 8 divided by 6.  And then we will subtract 3 divided by 6 from both sides, and get SV equal to 5 divided by 6. With that, we get our solution, which is E, 5 divided by 6.

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