Find Side AB: Calculate Using m & n Ratios

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In summary, the conversation discusses a problem involving a triangle and a line parallel to one of its sides. The goal is to find the ratio between two line segments, but there is uncertainty about whether there is enough information given in the problem. Several methods are proposed, including using similar triangles and the intercept theorem, but it is concluded that the problem may be more complicated than originally thought. The conversation also includes a correction to a misinterpretation of the given information.
  • #1
quee
7
2
Hello.
There is a problem:
"Through D on a side AB of the triangle ABC drawn a line parallel to AC intersecting BC in E. D is such that CD:DB = m:n. Find DE:AC"

1.png


So it is easy to find out that DB:AB equals to DE:AC as DE and AC are parallel. Since DB = n, there is only one need to express AB in terms of n and m. But how to do it? I tried using similar triangles, but I can't get AB through it. The trapezoid ADEC also gives no results as the second diagonal is unknown.
 
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  • #2
I think you need also ##AE## and the intersection of both diagonals ##AE## and ##CD## in, say ##F##. This will make things more complicated as there will be more lengths involved, but I don't see another way to get hold on ##CD##. Then use the intercept theorem.
 
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  • #3
fresh_42 said:
I think you need also ##AE## and the intersection of both diagonals ##AE## and ##CD## in, say ##F##.
Thank you. Actually, I considered that there may be not enough information in the problem. Now I am convinced.
 
  • #4
quee said:
Thank you. Actually, I considered that there may be not enough information in the problem. Now I am convinced.
I am not sure. It depends on how the result has to look like. E.g. I got for the quotient
$$\dfrac{DE}{AC} =\dfrac{n}{m} \cdot \dfrac{CD}{AD+DB}$$
but it's not clear whether this will do or not. I don't see a second equation for ##CD##.
 
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  • #5
Since ##DE## is parallel to ##AC##, you can use the similarity of triangles ##ABC## and ##ADE##. You have from the proportion that is given:
$$AD:DB = m:n \Rightarrow nAD = mDB$$
This means that ##AB = AD + DB = \left(\frac{m}{n} + 1\right) DB##. So now you have the proportion between ##AB## and ##DB## and from similarity you get the proportion between ##DE## and ##AC##.
 
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  • #6
Antarres said:
Since ##DE## is parallel to ##AC##, you can use the similarity of triangles ##ABC## and ##ADE##. You have from the proportion that is given:
$$AD:DB = m:n \Rightarrow nAD = mDB$$
This means that ##AB = AD + DB = \left(\frac{m}{n} + 1\right) DB##. So now you have the proportion between ##AB## and ##DB## and from similarity you get the proportion between ##DE## and ##AC##.
Did you misread the given information?

Statement says: ##CD:DB = m:n##, not ##AD:DB ##.
quee said:
D is such that CD:DB = m:n. Find DE:AC"
 
  • #7
@SammyS Indeed, I completely misread it. Apologies to the OP. In that case the exercise seems more complicated.
Might be that the method fresh gave would work, I so far don't see a clear way to get the proportion.
 
  • #8
Antarres said:
Indeed, I completely misread it. Apologies to the OP.

It is okay. Thank you for your reply.
If only $$AD:DB= m:n,$$ it would be completely obvious that $$\frac{DE}{AC} = \frac{n}{n+m}$$
But this is, unfortunately, a much harder problem. Or it may be an actual lack of some information in the problem.
 
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Related to Find Side AB: Calculate Using m & n Ratios

1. How do I find the value of side AB using m and n ratios?

To find the value of side AB, you will need to use the m and n ratios provided in the problem. These ratios represent the relationship between the sides of the triangle. You will then use the given measurements of the other sides to set up a proportion and solve for the missing value of side AB.

2. Can I use any values for m and n ratios to find the value of side AB?

No, you cannot use any values for the m and n ratios. These ratios must be provided in the problem and represent the specific relationship between the sides of the triangle. Using incorrect ratios will result in an incorrect value for side AB.

3. If I am given a right triangle, can I still use m and n ratios to find the value of side AB?

Yes, you can still use m and n ratios to find the value of side AB in a right triangle. The ratios will represent the relationship between the sides of the triangle, regardless of whether it is a right triangle or not.

4. What if I am missing one of the ratios, can I still find the value of side AB?

No, you will need both m and n ratios to find the value of side AB. If one of the ratios is missing, you will not have enough information to set up a proportion and solve for the missing value.

5. Are there any special steps I need to take when using m and n ratios to find the value of side AB?

Yes, you will need to make sure that the ratios are in the same units. If they are not, you will need to convert them to the same unit before setting up the proportion. Additionally, make sure to label your proportions correctly and double-check your calculations to ensure accuracy.

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