Longest Distance

Geometry Level 4

In the image, four squares of side length 2 are placed in the corners of a square of side length 6. Each of the points $W$ , $X$ , $Y$ , and $Z$ is a vertex of one of the small squares. Square $ABCD$ can be constructed with sides passing through $W$ , $X$ , $Y$ , and $Z$ .

The maximum possible distance from $A$ to $P$ is closest to which of the following numbers?

Problem and Image Source: CEMC (Past Contests).

4.0 5.8 5.0 5.4 5.2 6.0

1 solution

Hana Wehbi
Dec 12, 2016

Since $\angle WAX = 90^o$ regardless of the position of square $ABCD$ , then $A$ always lie on the semi-circle with diameter $WX$ .

The center of this semi-circle is the midpoint , $M$ , of $WX$ .

To get from $P$ to $M$ , we must go up $4$ units and to the left $3$ units since $WX = 2$ , so $PM^2= 3^2+4^2 = 25$ or $PM=5$ .

Since the semi-circle with diameter $WX$ has diameter $2$ , it has radius $1$ , so $AM=1$ and $MP=5$ .

Therefore, the maximum possible length of $AP$ is $5+1 = 6$ , when $A, M, P$ lie on a straight line.

A cleverly constructed problem. I solved it exactly as you anticipated. Thank you! (-:

Paul Hindess - 4 years, 6 months ago

I am glad you liked it :)

Hana Wehbi - 4 years, 6 months ago

Question, why must $WX$ , why not others ?

Jason Chrysoprase - 4 years, 6 months ago

I know, it doesn't look clear from the figure. I am going to draw it again later on today. But we are taking the maximum distance from A to P and that segment is going to be through the midpoint of $WX$ which is $M$ .

Hana Wehbi - 4 years, 6 months ago

By symmetry, the maximum distance could just as well be through the midpoint of WZ. Perhaps that was Jason's point?

[As an aside, I've just spent a good while trying to come up with a variation on this problem that also has a surprisingly nice (probably integer) solution. Turns out it isn't that easy to do - which makes the small numbers and diagram-simplicity of this problem seem all the more remarkable! Would love to know how you came to create this problem...]

Paul Hindess - 4 years, 6 months ago

@Paul Hindess – No, i mean the other side of $WX$

Jason Chrysoprase - 4 years, 6 months ago

Longest Distance

Problem and Image Source: CEMC (Past Contests).

1 solution

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