Asked in: March 2025 board paperIn the following figure DE || BC, then: (i) If DE = 4 cm, BC = 8 cm, \(A(\Delta ADE) = 25\) cm\(^2\), find \(A(\Delta ABC)\). (ii) If DE : BC = 3 : 5, then find \(A(\Delta ADE) : A(\square DBCE)\).[Figure: \(\Delta ABC…

Answer: (i) \(A(\Delta ABC) = 100\) cm\(^2\) (ii) 9 : 16

In \(\Delta ADE\) and \(\Delta ABC\): \(\angle A\) is common and \(\angle ADE \cong \angle ABC\) (corresponding angles, DE || BC). \(\therefore \Delta ADE \sim \Delta ABC\) (AA test).

(i) \(\dfrac{A(\Delta ADE)}{A(\Delta ABC)} = \dfrac{DE^2}{BC^2} = \dfrac{4^2}{8^2} = \dfrac{1}{4}\) \(\therefore \dfrac{25}{A(\Delta ABC)} = \dfrac{1}{4}\) \(\therefore A(\Delta ABC) = 100\) cm\(^2\).

(ii) \(\dfrac{A(\Delta ADE)}{A(\Delta ABC)} = \dfrac{DE^2}{BC^2} = \dfrac{9}{25}\). Let \(A(\Delta ADE) = 9k\) and \(A(\Delta ABC) = 25k\).

\(A(\square DBCE) = A(\Delta ABC) - A(\Delta ADE) = 25k - 9k = 16k\).

\(\therefore A(\Delta ADE) : A(\square DBCE) = 9k : 16k = 9 : 16\).