Simplify the fraction on the left side: (6)/(3) = 2. Then multiply both sides by x to get 2x = 9, which yields x = 4.5 cm.

In the given figure, find the value of $x$ , if $DE \parallel BC$ . Figure description: A triangle $ABC$ with a line segment $DE$ intersecting side $AB$ at point $D$ and side $AC$ at point $E$ , such that $DE \parallel BC$ . Given lengths are: - $AD = 6\text{ cm}$ - $DB = 3\text{ cm}$ - $AE = 9\text{ cm}$ - $EC = x$

Answer: $x = 4.5\text{ cm}$

Step-by-step solution

Step 1: State the Basic Proportionality Theorem

According to the Basic Proportionality Theorem (Thales' Theorem), if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. Since $DE \parallel BC$ in $\triangle ABC$ , the ratio $\frac{AD}{DB}$ must equal $\frac{AE}{EC}$ .

Step 2: Substitute the given values

Substitute the given measurements into the proportionality equation: $AD = 6\text{ cm}$ , $DB = 3\text{ cm}$ , $AE = 9\text{ cm}$ , and $EC = x$ .

Step 3: Solve for x

Simplify the fraction on the left side: $\frac{6}{3} = 2$ . Then multiply both sides by $x$ to get $2x = 9$ , which yields $x = 4.5\text{ cm}$ .

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