Question

2
dee is obtained by enlarging abc. if the area of abc is 3 square units, what is the area of dbe?
ba = 2
bd = 6
a. 27 square units
b. 24 square units
c. 12 square units
d. 9 square units

Explanation:

Step1: Find the ratio of side lengths

The ratio of \(BD\) to \(BA\) is \(\frac{BD}{BA}=\frac{6}{2} = 3\). Since \(\triangle ABC\) and \(\triangle DBE\) are similar (by AA similarity, as \(\angle ABC=\angle DBE\) and the sides are in proportion), the ratio of their side lengths \(k = 3\).

Step2: Use the area - ratio formula for similar triangles

For similar triangles, if the ratio of side lengths is \(k\), the ratio of their areas is \(k^{2}\). Let \(A_{1}\) be the area of \(\triangle ABC\) and \(A_{2}\) be the area of \(\triangle DBE\). The formula is \(\frac{A_{2}}{A_{1}}=k^{2}\). Given \(A_{1} = 3\) and \(k = 3\), we substitute into the formula: \(A_{2}=A_{1}\times k^{2}\).

Step3: Calculate the area of \(\triangle DBE\)

Substitute \(A_{1}=3\) and \(k = 3\) into \(A_{2}=A_{1}\times k^{2}\). Then \(A_{2}=3\times3^{2}=3\times9 = 27\).

Answer:

A. 27 square units