Cross-multiply the denominators and numerators to clear fractions. Expanding the left side gives x² + x, and expanding the right side gives (x - 1)(x + 3) = x² + 3x - x - 3 = x² + 2x - 3.

In the given figure, $DE \parallel BC$ . Find the value of $x$ . Figure Details: - A triangle $\triangle ABC$ with a line segment $DE$ parallel to base $BC$ , where point $D$ lies on side $AB$ and point $E$ lies on side $AC$ . - Lengths of segments: - $AD = x + 1$ - $DB = x - 1$ - $AE = x + 3$ - $EC = x$

Answer: 3

Step-by-step solution

Step 1: Apply Basic Proportionality Theorem

In $\triangle ABC$ , since line segment $DE$ is parallel to side $BC$ , by the Basic Proportionality Theorem (Theorem 6.1, Thales' Theorem), it divides sides $AB$ and $AC$ in the same ratio.

Step 2: Substitute the given segment lengths

Substitute the expressions $AD = x + 1$ , $DB = x - 1$ , $AE = x + 3$ , and $EC = x$ into the proportionality equation.

Step 3: Cross-multiply and expand the terms

Cross-multiply the denominators and numerators to clear fractions. Expanding the left side gives $x^2 + x$ , and expanding the right side gives $(x - 1)(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3$ .

Step 4: Solve for x

Subtract $x^2$ from both sides, which cancels the quadratic terms. Subtracting $2x$ from both sides yields $-x = -3$ , which simplifies directly to $x = 3$ .

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