Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem20

Levels 9&10 Video Solutions 2010 problem20

Video Transcription

Question number 20. A paper triangle was folded along the dashed line to obtain the figure on the far right. The area of the triangle before folding was 1.5  times larger than the area of the new figure and the sum of the areas of the three shaded regions is equal to 1. What was the area of the original triangle?  So let's label some regions over here. The shaded regions we can call them A, B, and C and then we know that A plus B plus C is equal to 1. We also know that  the area of the original triangle, so let's say area of triangle, is equal to 1.5 times the area of the new figure.  So let's see if we can get a better handle on these areas. What we have on the right-hand side that's not labeled is the white region so let's call that  X and if we were to unfold the triangle and keep track of the shaded triangles we would have a triangle maybe here like that, here like that, and then possibly  another one here and let me shade those back in with the same color so we have these three shaded regions X is right here and then by symmetry there is  another region X. So the area of the triangle is 2X plus again A, B is now over here, and C. So we can write that down in our equation the area of the  triangle on the left-hand side would be A plus B plus C plus 2X and that's equal to 1.5 the area of the new figure which is A plus B plus C and then plus just  1X because it's folded over on itself and then this we can solve. We know that A plus B plus C is equal to 1 so on the left-hand side we have 1 plus 2X on the  right-hand side we have 1.5 times 1 plus X and then we can solve this for X so 1 plus 2X is equal to 1.5 plus 1.5X 2X minus 1.5X is then equal to 1.5 minus 1  so 0.5X is equal to 0.5 and finally we obtain that X is equal to 1. Now to answer the question which is what is the area of the original triangle we go back  to our original formulation here we have A plus B plus C plus 2X and now we know each of those quantities so triangle has the area A plus B plus C plus 2X this is  equal to 1 and so that's 1 plus 2X or 1 plus 2 times 1 which gives us 3 so that's the area of the original triangle and so which is B as the answer

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