ABCD is a parallelogram. E is a point on AB such that 234×AE=EB. Let DE intersect AC at F. What is the ratio AC:AF?
To find the ratio AC:AF, we first need to determine the relationship between the lengths of the different line segments in the given parallelogram.
Let's begin by considering the given condition: 234 × AE = EB. This equation suggests that EB is three times longer than AE. In other words, AE takes one-fourth of the length of AB, while EB takes three-fourths of the length of AB. Therefore, if we divide AB into four equal parts, AE will mark the first part, and EB will cover the next three parts, as shown below:
A --- AE --- E --- EB --- B
Now, let's label the intersection point of DE and AC as F:
A --- E --- B
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D --- F --- C
Since ABCD is a parallelogram, we know that AD is parallel to BC and AB is parallel to CD. Therefore, we can deduce that triangles ADE and EFB are similar triangles (because they share an angle and have parallel sides).
Now, let's examine the ratio between the sides of these triangles. Since AE is one-fourth of AB, and EB is three-fourths of AB, the ratio between AE and EB can be expressed as 1:3. Additionally, since triangles ADE and EFB are similar, the ratio of corresponding sides will be the same. Thus, the ratio of DE to EF will also be 1:3.
Since DE intersects AC at F, we can relate the ratios of the corresponding sides. If we combine the ratios DE:EF and AC:CF, we can express the relationship between all four line segments as DE:EF:AC:CF, which is 1:3:AC:CF.
Now, we can solve for the ratio AC:AF. Since F is the intersection point of DE and AC, CF is equal to AF. Therefore, we have:
DE:EF:AC:CF = 1:3:AC:AF
Since DE:EF = 1:3, and DE:EF:AC:CF = 1:3:AC:CF, we can simplify the equation:
1:3:AC:CF = 1:3:AC:AF
From this equation, we can see that AC is equal to AC, so we can disregard it, which leaves us with:
1:3:CF = 1:3:AF
Since CF is equal to AF, we can substitute CF with AF:
1:3:AF = 1:3:AF
The ratio AF:AF is identical, so the simplified result is:
1:3
Therefore, the ratio AC:AF is 1:3.