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18 Aug 2021, 08:00
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Difficulty:
95%
(hard)
Question Stats:
43% (02:25) correct
57%
(02:26)
wrong
based on 251
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In the figure shown above, what is the area of the shaded hexagon ?
A. 140
B. 137.5
C. 127.5
D. 120
E. 110
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Attachment:
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23 Nov 2021, 08:10
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Official Solution:
In the figure shown above, what is the area of the shaded hexagon ?
A. \(140\)
B. \(137.5\)
C. \(127.5\)
D. \(120\)
E. \(110\)
The key to solve this question is to notice that the sides of the triangle are \(3*5=15\), \(4*5=20\) and \(5*5=25\). Any triangle whose sides are in the ratio 3:4:5 is a right triangle: \(15^2+20^2=25^2\)
Re-draw:
Notice that the small grey triangles, in top and right corners, are similar to our large triangle (all their angles are equal), so the ratio of the sides in grey triangles will also be 3:4:5. And since the hypotenuse of these grey triangles equal to 5, then both of these triangles are 3:4:5 triangles. This means that horizontal leg of the top left grey triangle will be 4 and the vertical leg of the bottom right grey triangle will be 3.
The area of the shaded hexagon will be (the area of the large triangle) - (the areas of three smaller triangles in the corners) = \(\frac{15*20}{2}-(\frac{4*5}{2}+\frac{3*5}{2} +\frac{5*5}{2})=120\).
Answer: D
In the figure shown above, what is the area of the shaded hexagon ?
A. \(140\)
B. \(137.5\)
C. \(127.5\)
D. \(120\)
E. \(110\)
The key to solve this question is to notice that the sides of the triangle are \(3*5=15\), \(4*5=20\) and \(5*5=25\). Any triangle whose sides are in the ratio 3:4:5 is a right triangle: \(15^2+20^2=25^2\)
Re-draw:
Notice that the small grey triangles, in top and right corners, are similar to our large triangle (all their angles are equal), so the ratio of the sides in grey triangles will also be 3:4:5. And since the hypotenuse of these grey triangles equal to 5, then both of these triangles are 3:4:5 triangles. This means that horizontal leg of the top left grey triangle will be 4 and the vertical leg of the bottom right grey triangle will be 3.
The area of the shaded hexagon will be (the area of the large triangle) - (the areas of three smaller triangles in the corners) = \(\frac{15*20}{2}-(\frac{4*5}{2}+\frac{3*5}{2} +\frac{5*5}{2})=120\).
Answer: D
General Discussion
18 Aug 2021, 08:18
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The given triangle is a Pythagorean triplet of 3:4:5.
Therefore it is a right-angled triangle right angled at C.
Area of ABC = 1/2*20*15 = 150
Area of the smaller triangle with a vertice at C is = 1/2*5*5 = 12.5
So the area of the remaining section is 150-12.5 = 137.5
Options A and B are gone.
The maximum area of the two smaller traingles at vertices A and B can be 5*5 = 25
Hence the min area of the shaded region has to be = 137.5 - 25 = 112.5 [Option E eliminated]
Option C = 127.5 is unrealistic since the triangles can't add up to 10 only.
Hence the answer is option D.
Therefore it is a right-angled triangle right angled at C.
Area of ABC = 1/2*20*15 = 150
Area of the smaller triangle with a vertice at C is = 1/2*5*5 = 12.5
So the area of the remaining section is 150-12.5 = 137.5
Options A and B are gone.
The maximum area of the two smaller traingles at vertices A and B can be 5*5 = 25
Hence the min area of the shaded region has to be = 137.5 - 25 = 112.5 [Option E eliminated]
Option C = 127.5 is unrealistic since the triangles can't add up to 10 only.
Hence the answer is option D.
