Crab, Crow, & Snake

Geometry Level 5

A crab is walking sideways from one field hole to another. Upon reaching the midpoint between the 2 holes ( $B$ and $C$ ), the crab realizes that it is caught in the line of 2 predators: the crow and the snake.

As shown, the crow is 21 feet away from the left hole at $B$ , while the snake is 17 feet from the right hole at $C$ . Also, both predators are 72 feet apart from each other. From the crab's standing point $P$ , all the lengths (in feet) towards the holes and other animals have integer values.

How far apart (in feet) are the crab's holes?

Note: The figure is not drawn to scale.

The answer is 62.

2 solutions

Jeremy Galvagni
Dec 8, 2018

Let's not worry about the fact that a crab knows these distances are integers. Since there are vertical angles making the triangles have an equal angle we can use the Law of Cosines SSS version to make two equal fractions. Call $AP=x$ and $BP=CP=y$ so $PD=(72-x)$ and we have

$\frac{x^{2}+y^{2}-21^{2}}{2xy} = \frac{(72-x)^{2}+y^{2}-17^{2}}{2(72-x)y}$

Solve this for y

$\large \sqrt{\frac{-x^{3}+108x^{2}-2227x-15876}{x-36}}$

Making a table we can find this equation has five integer solutions $0<x<72$ but most don't work

x	y	comment
17	38	PDC degenerate
34	55	ABP degenrate
38	17	ABP degenrate
40	31	works
55	34	ABP degenrate

So the only solution that fits the problem is $y=31$ and so $BC=2y=\boxed{62}$ feet.

Other than iterative method, is there a way to get integer values for y?

Satyen Dhamankar - 2 years, 5 months ago

Vinod Kumar
Dec 27, 2018

Solved Diophantine equation (using WolframAlpha) given by equating the cosine of the angle of triangles in terms of three side lengths and obtained two triangles as (17,31,32) and (21,31,40) and the common equal angle as CosInv (53/62).

Answer=62

Crab, Crow, & Snake

The answer is 62.

2 solutions

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