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Geometry and Trigonometry / Lines, angles, and triangles Difficulty: Medium

Two nearby trees are perpendicular to the ground, which is flat. One of these trees is $10$ feet tall and has a shadow that is $5$ feet long. At the same time, the shadow of the other tree is $2$ feet long. How tall, in feet, is the other tree?

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Explanation

Choice B is correct. Each tree and its shadow can be modeled using a right triangle, where the height of the tree and the length of its shadow are the legs of the triangle. At a given point in time, the right triangles formed by two nearby trees and their respective shadows will be similar. Therefore, if the height of the other tree is $x$ , in feet, the value of $x$ can be calculated by solving the proportional relationship $\frac{10 feet tall}{5 feet long} = \frac{x feet tall}{2 feet long}$ . This equation is equivalent to $\frac{10}{5} = \frac{x}{2}$ , or $2 = \frac{x}{2}$ . Multiplying each side of the equation $2 = \frac{x}{2}$ by $2$ yields $4 equals x$ . Therefore, the other tree is $4 feet$ tall.

Choice A is incorrect and may result from calculating the difference between the lengths of the shadows, rather than the height of the other tree.

Choice C is incorrect and may result from calculating the difference between the height of the $10$ -foot-tall tree and the length of the shadow of the other tree, rather than calculating the height of the other tree.

Choice D is incorrect and may result from a conceptual or calculation error.