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Can someone help me prove this geometry problem?
Two straight lines parallel to the base of a trapezoid divide each lateral side into three equal parts. The entire trapezoid is separated by the lines into three parts. Find the area of the middle part if the areas of the upper and lower part are S1 and S2, respectively.
The answer is 0.5(S1+S2). Help will be much appreciated.
2 Answers
- Anonymous4 years ago
Let the lengths of the || sides from smallest to largest be x, y, z and w respectively, and h be the distance between consecutive || sides. Let S3 be the area of the middle part.
y = 0.5(x + z)
z = 0.5(y + w)
S1 = 0.5(x + y)h
S2 = 0.5(z + w)h
S3 = 0.5(y + z)h
= 0.5[0.5(x + z) h+ 0.5(y + w)h]
= 0.5[0.5(x + y)h + 0.5(z + w)h]
= 0.5(S1 + S2).
- PuzzlingLv 74 years ago
It's probably easiest to see if you make a second copy of the trapezoid, flip it upside down and place it next to the first one. The resulting shape is made up of 3 parallelograms, each the same area.
So we can say:
S1 + S3 = S2 + S2
From there it is pretty simple to get what you want:
2(S2) = S1 + S3
S2 = ½(S1 + S3)
