In △ABC , points D and E are on the sides BC and AC, respectively. BE and AD intrested at point T such that AD:AT=4:3, and BE:BT=5:4. Point F lies on AC such that DF is parallel to BE. Then, BD:CD is

The correct option is 11:4.

To find the ratio of BD to CD, we can use the properties of similar triangles formed by parallel lines. Let's break down the given information step-by-step.

Step 1: Understand the given ratios on the intersecting segments.

We are given the ratio $\mathrm{AD}:\mathrm{AT}=4:3$. This tells us that if the total length of AD is 4 parts, the length of AT is 3 parts. The remaining part, TD, would be $4-3=1$ part.

Therefore, we can write the ratio of AT to AD as:

$\frac{\mathrm{AT}}{\mathrm{AD}}=\frac{3}{4}$

Similarly, we are given the ratio $\mathrm{BE}:\mathrm{BT}=5:4$. This means if BE is 5 parts, BT is 4 parts. The remaining segment, TE, is $5-4=1$ part.

So, the ratio of TE to the total length BE is:

$\frac{\mathrm{TE}}{\mathrm{BE}}=\frac{1}{5}$

Step 2: Use the parallel lines to find similar triangles.

The problem states that point F lies on AC such that DF is parallel to BE. Because the line segment TE is just a section of the entire line segment BE, it must also be true that TE is parallel to DF.

Now, let's look at the triangle ADF. Since TE is parallel to the base DF, triangle ATE is similar to triangle ADF.

Because these triangles are similar, the ratio of their corresponding sides must be equal. Therefore:

$\frac{\mathrm{TE}}{\mathrm{DF}}=\frac{\mathrm{AT}}{\mathrm{AD}}$

Substituting the ratio we found earlier:

$\frac{\mathrm{TE}}{\mathrm{DF}}=\frac{3}{4}$

We can rearrange this to express DF in terms of TE:

$\mathrm{DF}=\frac{4}{3}\mathrm{TE}$

Step 3: Connect DF to BE.

From Step 1, we know that $\mathrm{TE}=\frac{1}{5}\mathrm{BE}$. Let's substitute this into our equation for DF:

$\mathrm{DF}=\frac{4}{3}\times \left(\frac{1}{5}\mathrm{BE}\right)$

$\mathrm{DF}=\frac{4}{15}\mathrm{BE}$

This gives us the ratio of DF to BE:

$\frac{\mathrm{DF}}{\mathrm{BE}}=\frac{4}{15}$

Step 4: Find the final ratio in triangle CBE.

Next, let's focus on triangle CBE. We constructed DF to be parallel to BE. Because DF is parallel to the base BE of triangle CBE, triangle CDF is similar to triangle CBE.

Again, using the property of similar triangles, the ratio of their corresponding sides must be equal. Thus:

$\frac{\mathrm{CD}}{\mathrm{CB}}=\frac{\mathrm{DF}}{\mathrm{BE}}$

Substituting the ratio we just found:

$\frac{\mathrm{CD}}{\mathrm{CB}}=\frac{4}{15}$

This means if the length of CD represents 4 parts, the total length of CB represents 15 parts.

Since point D lies on the line segment CB, the total length CB is the sum of CD and BD:

$\mathrm{CB}=\mathrm{CD}+\mathrm{BD}$

Substituting our part values:

$15=4+\mathrm{BD}$

$\mathrm{BD}=15-4=11$

Therefore, the length of BD is 11 parts and the length of CD is 4 parts. The ratio of BD to CD is exactly:

$\mathrm{BD}:\mathrm{CD}=11:4$