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D, E and F are midpoints of the sides of ABC as shown
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Updated on: 27 Jun 2024, 22:26
Updated on: 27 Jun 2024, 22:26
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D, E, and F are midpoints of the sides of \(\triangle ABC\) as shown
Quantity A |
Quantity B |
The sum of the areas of the shaded regions |
The area of the region enclosed by quadrilateral \(DBEF\) |
Big book test 7 section 4 #7
How do i solve this I have no idea. Please help.
Re: D, E and F are midpoints of the sides of ABC as shown
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27 Jun 2024, 22:27
27 Jun 2024, 22:27
Expert Reply
Join the points D and E as shown in the diagram below:
Now DEAF is a paralleogram and DF is a diagonal. Since lined passing through midpoint of two sides of a triangle is paralle to the third side. So DE is paralle to AC and DF is parallel to BC and so on.
So Area of triangle DAF (shaded triangle on the left)=area of the triangle DEF. A diagonal of a parallelogram divides it into two triangles of equal area.
Similarly in parallelogram DECF the area of triangle DEF= ECF. Same logic as above. Hence the two shaded areas are equal and also equal to the unshaded triangle DEF.
Now the final paralleogram BDFE we have triangle BDE and triangle FDE. They are also equal in area as DE is diagonal of paralleogram BDEF.
Hence all 4 areas of the above triangle are equal. Thus the shaded area is equ\l to the unshaded area.
Hence option C is the best fit!
Now DEAF is a paralleogram and DF is a diagonal. Since lined passing through midpoint of two sides of a triangle is paralle to the third side. So DE is paralle to AC and DF is parallel to BC and so on.
So Area of triangle DAF (shaded triangle on the left)=area of the triangle DEF. A diagonal of a parallelogram divides it into two triangles of equal area.
Similarly in parallelogram DECF the area of triangle DEF= ECF. Same logic as above. Hence the two shaded areas are equal and also equal to the unshaded triangle DEF.
Now the final paralleogram BDFE we have triangle BDE and triangle FDE. They are also equal in area as DE is diagonal of paralleogram BDEF.
Hence all 4 areas of the above triangle are equal. Thus the shaded area is equ\l to the unshaded area.
Hence option C is the best fit!
Re: D, E and F are midpoints of the sides of ABC as shown
[#permalink]
27 Jun 2024, 22:28
27 Jun 2024, 22:28
Expert Reply
