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23 Nov 2021, 08:09
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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. \(110\)
23 Nov 2021, 08:09
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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
25 Feb 2026, 04:43
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