In ∆𝐴𝐵𝐶, 𝐷𝐸||𝐵𝐶, find the value of 𝑥.

In Triangle ABC DE is Parallel to BC - Find the Value of X

In the given triangle ABC, it is given that DE is parallel to BC. Our objective is to find the value of x. Since DE is parallel to BC, the triangles ∆ABC and ∆ADE are similar by the Basic Proportionality Theorem (BPT).

Step 1: Understanding the Similarity Rule

Since DE || BC, we can apply the similarity condition. According to the proportionality theorem:

AB/AD = AC/AE

Using the segment division property:

(AD + DB)/AD = (AE + EC)/AE

Simplifying both sides:

1 + DB/AD = 1 + EC/AE

Thus,

DB/AD = EC/AE

Step 2: Substituting Known Values

In the given triangle ABC, we know:

• AD = x
• DB = x + 1
• AE = x + 3
• EC = x + 5

Now substitute these values into the proportionality equation:

(x + 1)/x = (x + 5)/(x + 3)

Step 3: Solving the Equation

Cross multiplying both sides:

x(x + 5) = (x + 3)(x + 1)

Expanding both sides:

x² + 5x = x² + x + 3x + 3

Simplifying further:

x² + 5x = x² + 4x + 3

Rearranging terms:

5x - 4x = 3

Thus,

x = 3

Step 4: Final Answer

Therefore, in the given triangle ABC where DE is parallel to BC, the value of x is 3.