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?
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.
+396 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 \(16\pi.\)
