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Algebra / Linear equations in two variables Difficulty: Medium

A total of $2$ squares each have side length $r$ . A total of $6$ equilateral triangles each have side length $t$ . None of these squares and triangles shares a side. The sum of the perimeters of all these squares and triangles is $210$ . Which equation represents this situation?

$6 r + 24 t = 210$

$2 r + 6 t = 210$

$8 r + 18 t = 210$

$6 r + 2 t = 210$

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Explanation

Choice C is correct. It’s given that a total of $2$ squares each have side length $r$ . Therefore, each of the squares has perimeter $4 r$ . Since there are a total of $2$ squares, the sum of the perimeters of these squares is $4 r plus 4 r$ , which is equivalent to $2 left parenthesis 4 r right parenthesis$ , or $8 r$ . It’s also given that a total of $6$ equilateral triangles each have side length $t$ . Therefore, each of the equilateral triangles has perimeter $3 t$ . Since there are a total of $6$ equilateral triangles, the sum of the perimeters of these triangles is $3 t + 3 t + 3 t + 3 t + 3 t + 3 t$ , which is equivalent to $6 left parenthesis 3 t right parenthesis$ , or $18 t$ . Since the sum of the perimeters of the squares is $8 r$ and the sum of the perimeters of the triangles is $18 t$ , the sum of the perimeters of all these squares and triangles is $8 r plus 18 t$ . It’s given that the sum of the perimeters of all these squares and triangles is $210$ . Therefore, the equation $8 r + 18 t = 210$ represents this situation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

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