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What is the ratio of the area of a rectangular TV screen with a diagonal of 18 inches to that of a rectangular screen with a diagonal of 15 inches?
(1) Both screens have the same width-to-length ratio.
(2) The width of the 18-inch screen is 20% greater than the width of the 15-inch screen.
M10-05
(1) Both screens have the same width-to-length ratio.
(2) The width of the 18-inch screen is 20% greater than the width of the 15-inch screen.
M10-05
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27 May 2010, 05:58
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Official Solution:
What is the ratio of the area of a rectangular TV screen with a diagonal of 18 inches to that of a rectangular screen with a diagonal of 15 inches?
In two similar triangles (or rectangles), the ratio of their areas is the square of the ratio of their corresponding sides.
Given: ratio of diagonals \(= \frac{18}{15}=1.2\) (note that diagonals act as the hypotenuses in right triangles formed by width and length).
(1) Both screens have the same width-to-length ratio.
This implies that the rectangles are similar. Consequently, the right triangles formed by the diagonals are similar as well, which means that the ratio of the areas of the triangles equals the ratio of the areas of the rectangles \(= (\frac{18}{15})^2 = 1.44\). Sufficient.
(2) The width of the 18-inch screen is 20% greater than the width of the 15-inch screen.
Given: the ratio of widths = 1.2 = ratio of diagonals (hypotenuse). Thus, the right triangles formed by the width and length are similar (in right triangles, if two corresponding sides have the same ratio, in our case \(\frac{W}{w}=\frac{D}{d}=1.2\), then these right triangles are similar). Therefore, the rectangles are similar as well, which means that the ratio of the areas of the triangles equals the ratio of the areas of the rectangles \(= (\frac{18}{15})^2=1.44\). Sufficient.
Answer: D
What is the ratio of the area of a rectangular TV screen with a diagonal of 18 inches to that of a rectangular screen with a diagonal of 15 inches?
In two similar triangles (or rectangles), the ratio of their areas is the square of the ratio of their corresponding sides.
Given: ratio of diagonals \(= \frac{18}{15}=1.2\) (note that diagonals act as the hypotenuses in right triangles formed by width and length).
(1) Both screens have the same width-to-length ratio.
This implies that the rectangles are similar. Consequently, the right triangles formed by the diagonals are similar as well, which means that the ratio of the areas of the triangles equals the ratio of the areas of the rectangles \(= (\frac{18}{15})^2 = 1.44\). Sufficient.
(2) The width of the 18-inch screen is 20% greater than the width of the 15-inch screen.
Given: the ratio of widths = 1.2 = ratio of diagonals (hypotenuse). Thus, the right triangles formed by the width and length are similar (in right triangles, if two corresponding sides have the same ratio, in our case \(\frac{W}{w}=\frac{D}{d}=1.2\), then these right triangles are similar). Therefore, the rectangles are similar as well, which means that the ratio of the areas of the triangles equals the ratio of the areas of the rectangles \(= (\frac{18}{15})^2=1.44\). Sufficient.
Answer: D
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