Ratios of Line Segments in Parallelogram ABCD: Divide BC & AE

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In summary, the problem involves a parallelogram ABCD with DC extended to E in a ratio of 3:2. Line AE intersects BC at F, and the goal is to determine the ratios in which F divides BC and AE. The division-point theorem may be useful, but the given ratio of 3:2 for DE:EC does not seem to match. The concept of external division of a point is also mentioned, but it is unclear how it applies in this case.
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Homework Statement


in the parallelogram ABCD, DC is extended to E so that DE:EC= 3:2. The line AE meets BC at F. Determine the ratios in which F divides BC and F divides AE

The Attempt at a Solution



so i figured i could use the division-point theorem

OP=[b/(a+b)]*OA +[a/(a+b)]OB

but i don't know how to use this equation in this case..i really don't know how to get started T-T
 
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I'm not sure I understand, ED < EC so how can the ratio be 3:2?? I am familiar with external division of a point but are you sure that is it?
 
  • #3
I would approach this problem by first drawing a diagram of the parallelogram ABCD and its extended side DC. This will help visualize the given information and determine the unknown ratios.

Next, I would use the division-point theorem to find the ratio in which F divides BC. This theorem states that the ratio in which a point divides a line segment is equal to the ratio of the lengths of the two segments on either side of the point. In this case, the point F divides BC into two segments, BF and FC. Therefore, the ratio in which F divides BC is equal to the ratio of BF to FC.

To find the ratio of BF to FC, we can use the given information that DE:EC=3:2. Since DE is the extension of DC, it is equal to 3 times the length of DC. And since DC and EC are part of the same line, their lengths are directly proportional to the ratio 3:2. This means that DC is 3/5 of the total length of DC+EC, and EC is 2/5 of the total length. Therefore, DC is 3/5 of the total length of BC, and EC is 2/5 of the total length.

Now, we can use the division-point theorem to find the ratio of BF to FC. Plugging in the values we found, we get:

BF/FC = (3/5)/(2/5) = 3/2

So, the ratio in which F divides BC is 3:2.

To find the ratio in which F divides AE, we can use a similar approach. We know that the line AE meets BC at F, so the ratio in which F divides AE is equal to the ratio of AF to FE. And since DE:EC=3:2, we know that the ratio of AF to FE is also 3:2.

So, the ratios in which F divides BC and AE are both 3:2.
 

1. What is the formula for finding the ratio of line segments in a parallelogram?

The formula for finding the ratio of line segments in a parallelogram is AB:BC = AD:DC.

2. How do you divide line segments in a parallelogram?

To divide line segments in a parallelogram, you can use the formula AB/BC = AD/DC. This means that the ratio of the lengths of one pair of opposite sides is equal to the ratio of the lengths of the other pair of opposite sides.

3. Can the ratio of line segments in a parallelogram be greater than 1?

Yes, the ratio of line segments in a parallelogram can be greater than 1. This occurs when one side is longer than the other, resulting in a ratio greater than 1.

4. What is the relationship between the ratio of line segments and the properties of a parallelogram?

The ratio of line segments in a parallelogram is directly related to the properties of a parallelogram. This is because the ratio of line segments is determined by the equal opposite sides and opposite angles of a parallelogram.

5. Why is finding the ratio of line segments important in a parallelogram?

Finding the ratio of line segments in a parallelogram is important because it helps determine the properties of the parallelogram, such as the length of its sides and the measure of its angles. It can also be used to solve problems involving the sides and angles of a parallelogram.

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