What is the area of rectangle ABCD? (1) Diagonal AC is twice as long

Official Solution:


What is the area of rectangle \(ABCD\) ?

(1) The length of diagonal \(AC\) is twice the length of side \(CD\).

This statement alone is insufficient to determine the area of the rectangle. While it establishes a relationship between the diagonal and one side, it does not provide any specific dimensions of the rectangle.

(2) The length of diagonal \(AC\) is 0.1 meters longer than the length of side \(AD\).

Similarly, this statement alone is also insufficient to determine the area of the rectangle.

(1)+(2) From statement (1), we have \(CD = \frac{AC}{2}\). Since \(AC^2 = CD^2 + AD^2\), substituting the expression for \(CD\) gives \(AC^2 = (\frac{AC}{2})^2 + AD^2\), which simplifies to \(\sqrt{3}*AC = 2*AD\). From statement (2), we know that \(AC = AD + 0.1\). By combining these equations, \(\sqrt{3}*AC = 2*AD\) and \(AC = AD + 0.1\), we have two distinct linear equations with two unknowns, which can be solved to obtain the values of \(AC\) and \(AD\). With these values, we can then calculate \(CD\) and determine the area of the rectangle. Thus, the information from both statements is sufficient.


Answer: C