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16 Sep 2014, 01:09
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What is the area of rectangle \(ABCD\) ?
(1) The length of diagonal \(AC\) is twice the length of side \(CD\).
(2) The length of diagonal \(AC\) is 0.1 meters longer than the length of side \(AD\).
(1) The length of diagonal \(AC\) is twice the length of side \(CD\).
(2) The length of diagonal \(AC\) is 0.1 meters longer than the length of side \(AD\).
16 Sep 2014, 01:09
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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
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
08 Nov 2017, 11:47
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To all,
Combining both statements, is it safe to assume that the diagonal splits the rectangle into two, 30-60-90 triangles and that with information from statement 2 we can determine the area of each triangle?
Combining both statements, is it safe to assume that the diagonal splits the rectangle into two, 30-60-90 triangles and that with information from statement 2 we can determine the area of each triangle?
