Triangles \( \triangle D E F \) and \( \triangle H J K \) are similar. \( \triangle D E F \) has side lengths \( D E=10, E F= \) 15, and \( F D=20 \). Which of the following could be the side lengths of \( \triangle H J K \) ? A. \( 5,7,9 \) B. \( 2,6,7 \) C. \( 2,3,4 \) b. \( 8,10,12 \) D. 8
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Triangles are similar when their corresponding angles are equal and their side lengths are proportional. For triangles \( \triangle D E F \) with sides 10, 15, and 20, the ratios of the sides can be simplified as follows: The side lengths are in the ratio \( \frac{10}{10} : \frac{15}{10} : \frac{20}{10} \) or \( 1 : 1.5 : 2 \). Now, let’s evaluate the given options to see if any of these maintain a similar ratio: A set of side lengths that keeps the same proportional relationship as \( \triangle D E F \) would need one side to satisfy being half, one to be 1.5 times, and one to be double, but none of the options provided match this. Thus, you would need to check which sets of side lengths can be scaled from the \( 1 : 1.5 : 2 \). The only option that comes close to respecting the ratio while being whole numbers would need to be higher than any of the given sets. Hence none of the options conform to the required similarity based on the proportion derived from the original triangle.
