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Two points, ${}A$ and $B$, are on the pavement $6$ inches apart. Cassidy the caterpillar traces out a path walking along all the points that are twice as far from ${}A$ as from $B$. Cassidy has red paint on her feet, so she literally traces out the whole path! What is the area of the region inside Cassidy's path?

 May 11, 2023
 #1
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Label a point C at the midpoint of AB. Let D be a point on AB such that AD=2BD. Then the path traced by Cassidy is the rhombus ABCD. Let E be the foot of the perpendicular from D to AC. Then DE=DB=3 inches and AC=6 inches, so AE=3 inches and EC=3 inches. Therefore, the area of the rhombus is 1/2​(AC)(DE)=1/2​(6)(3)=9​ square inches.

 

The answer is 9 square inches.

 May 11, 2023
 #2
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Arrange for A to be at the origin of a co-ordinate system, B at the point (6, 0) and Cassidy at the point with co-ordinates (x, y) then 

\(\sqrt{x^{2}+y^{2}}=2\sqrt{(x-6)^{2}+y^{2}}, \)

so, squaring and tidying up,

\(x^{2}+y^{2}=4\{x^{2}-12x+36+y^{2}\}, \\ 3x^{2}+3y^{2}-48x+144=0,\\ x^{2}+y^{2}-16x+48=0, \\ (x-8)^{2}+y^{2}=16=4^{2}.\)

This is a circle with radius 4, so the area of the region is \(16\pi.\)

 May 11, 2023

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