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hi there welcome to we learn daily in
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this video we will find the area of the
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rectangle in the image if the areas of
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the other rectangles are 24 42 9 12 and
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18 square units as usual before watching
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try to solve this problem on your own
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now we'll go over an amazing method that
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allows even a fifth grader to quickly
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and easily solve any seemingly complex
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problem of this type let's start by
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considering the simplest case we'll take
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a rectangle and divide it into four
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smaller rectangles using lines parallel
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to its adjacent sides we'll label the
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areas of the smaller rectangles as S1 S2
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S3 and S4 will prove that the areas
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proportion S1 / S3 = S2 / S4 if we reite
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this in terms of the fundamental
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property of proportions we get that the
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product of the outer terms
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S1 * S4 must equal the product of the
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inner terms S2 * S3 let's quickly prove
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this we'll label the segments formed by
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the division of the rectangle as a b c
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and d then the area of the first
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rectangle is equal to B * C the area of
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the second rectangle is b multili d the
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area of the third is a multiplied C and
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the area area of the fourth is a * D
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this means that S1 * S4 = B * C * a * D
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and this product is the same as that of
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S2 * S3 thus the property of the areas
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is proven so when dividing any rectangle
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into Parts with two lines the product of
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the areas of the smaller rectangles
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along one diagonal equals the product of
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the areas along the other diagonal now
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we'll use this property to solve our
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problem first we'll extend the side of
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the small rectangle with an area of nine
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like this as a result we obtain a
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rectangle for which we can apply the
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area property we just proved if we
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denote the unknown area of the new small
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rectangle as s then according to the
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area property the product of the areas
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along one diagonal 18 * s must equal the
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product of the areas along the other
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diagonal 9 multiplied by 12 dividing
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both sides by 18 we get S = 99 * 12 / 18
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simplifying we get s = 6 therefore the
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area of the new small rectangle is 6 if
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we subtract this area from the rectangle
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with an area of 24 we get 24 - 6 which
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equals 18 this means the area of the
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remaining part of the rectangle is 18
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now we have another simple case of
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dividing the largest rectangle into four
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parts for which the area property holds
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the product of the are areas along one
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diagonal the yellow rectangles must
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equal the product of the areas along the
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other diagonal the blue rectangles
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taking one yellow rectangle we add 9 and
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18 to its area getting x + 9 + 18 and
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multiply this by 18 this must equal the
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product of the areas of the two blue
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rectangles 6 + 12 multiplied by 42 we
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combine like terms in the parentheses on
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the left side and get x + 27 * 18 and
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all of this should equal 6 + 12 which
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42 we then divide both sides by 18
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simplifying the equation we get x + 27 =
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42 solving for x we find that x = 42 -
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27 therefore x = 15 the area of the
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rectangle in the image is 15 square
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units the problem is solved if the
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solution is clear give it a like leave a
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comment and don't forget to subscribe to
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the channel thank you for watching and
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let's see you in the next video goodbye
