ABCD is a parallelogram and E is the midpoint of the side CD . Prove that area (ABED )=3×area of (triangle BEC )
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To prove that the area of quadrilateral ABED is three times the area of triangle BEC, we can use the concept of area ratios. Here's how:
1. Draw a diagram of the given parallelogram ABCD and mark the point E as the midpoint of side CD.
A _________ B
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E _________ D
2. Since E is the midpoint of CD, we know that CE = ED.
3. Let F be the point on line segment AB such that EF is parallel to BC. Connect EF.
A _________ B
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F | |
E _________ D
4. Since ABCD is a parallelogram, we know that BC is parallel to AD. Therefore, EF is also parallel to AD.
5. Now, we have two pairs of parallel lines: EF || BC and EF || AD. This implies that EFAB is also a parallelogram.
6. Since E is the midpoint of CD (given), it means that EF is the mid-segment of triangle CBD. The mid-segment of the triangle divides it into two equal areas.
7. The area of triangle BEC is half the area of triangle BCD. Therefore, area(BEC) = 0.5 * area(BCD).
8. It means that the area of triangle BCD is twice the area of triangle BEC. So, area(BCD) = 2 * area(BEC).
9. Since EF is the mid-segment of triangle CBD, it means that the area of triangle BCD is twice the area of triangle CEF. Therefore, area(BCD) = 2 * area(CEF).
10. Combining Steps 8 and 9, we have: 2 * area(BEC) = 2 * area(CEF).
11. Considering the parallelogram EFAB, which shares the base EF with triangle CEF, we can say that area(EFAB) = 2 * area(CEF). This is because the height of the parallelogram is the same as the height of triangle CEF.
12. So, we can rewrite the equation as area(BEC) = 0.5 * area(EFAB).
13. Now, let's focus on proving that the area of quadrilateral ABED is three times the area of triangle BEC.
14. Since ABED is also a parallelogram (opposite sides are parallel), we know that triangle BEC is equal in area to triangle AED.
15. So, area(ABED) = area(BEC) + area(AED) = area(BEC) + area(BEC) = 2 * area(BEC).
16. From Step 12, we have that area(BEC) = 0.5 * area(EFAB). Therefore, area(ABED) = 2 * (0.5 * area(EFAB)) = area(EFAB).
17. Finally, we conclude that the area of quadrilateral ABED is equal to the area of triangle BEC, which is three times the area of triangle BEC: area(ABED) = 3 * area(BEC).
By following the above steps, we have proven that the area of quadrilateral ABED is three times the area of triangle BEC.