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Fido's leash is tied to a stake at the center of his yard, which is in the shape of a an equilateral triangle.  His leash is exactly long enough to reach the midpoint of each side of his yard. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest radical form as sqrt(a)/b*pi, what is the value of the product ab?

 Apr 20, 2021
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careful with the expression sqrt(a)/b*pi. some people could thing it's $\dfrac{\sqrt{a}}{b\pi}$.

 

we can say that the length of one side of the yard is 2 meters(ignore how small it is). With numbers, it is much easier to calculate.

We can draw lines from the center of the triangle to the midpoints of the sides, to get 3 quadrilaterals. we can cut those in half so that there are lines going from the center to the vertices, so we have 6 congruent 30-60-90 triangles. We know that the long leg of the triangles is 1, so the radius of the area Fido's leash is able to reach is $\dfrac1{\sqrt3}$. So, the area Fido is able to go in is

$(\dfrac{1}{\sqrt3})^2\cdot\pi=\dfrac13\pi$

the area of the triangle is height x base / 2 = 2 $\sqrt3$ x 2 / 2 = 2$\sqrt3$

so the fraction of the yard Fido is able to cross is

$\dfrac{\dfrac13}{2\sqrt3}\pi=\dfrac{\sqrt3}{18}\pi$

so a=3 and b=18. therefore: ab=$\boxed{54}$

 Apr 20, 2021

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