Double Tag

Geometry Level 2

Find the length of the shortest path that can be drawn from the point $(6, 3)$ to the point $(2, 8)$ such that the path touches the $x$ -axis and the $y$ -axis once.

3\sqrt{5} + 2\sqrt{17}

\sqrt{185}

14

\sqrt{149}

2 solutions

Andrew Ellinor
Sep 29, 2015

We know the shortest distance between two points is a straight line. With this in mind, we start by reflecting $(2, 8)$ over the $x$ -axis and then over the $y$ -axis.

If you connect $(6, 3)$ to the newly reflected point $(-2, -8)$ to make a straight line and the distance traveled by this path is the same as the distance traveled by its reflection through the axes. Given that it's a straight line (and the shortest distance between two points is a straight line), this will give the shortest distance. The distance between $(6, 3)$ and $(-2, -8)$ is $\sqrt{(6 - (-2))^2 + (3 - (-8))^2} = \sqrt{185}$

Dear Andrew why can't it be extending line segment both the sides to to touch x and y axis. Point on x axis (8.4 ,0) and on yaxis (0,10.5).the distance BT them square root of 180.81.pl tell me flaw in this.

nagarjuna reddy - 5 years, 5 months ago

It doesn't technically answer the question. You're supposed to start at one point, then go to the other and end there. Your answer does create a path that connects the points and touches the x and y axis, but it starts off the points. If you were to extend it so that the point starts at (6,3), then goes to the x-axis, then follows your path, then goes to (2,8), it would be valid but longer than Andrew's path. Does this answer your question?

Austin Antonacci - 5 years, 4 months ago

How is the distance of straight line connecting (6,3) to (-2,-8) is equal to the required distance ?? I mean is it any observation or some theorem which states so ??

amey kashyap - 5 years, 1 month ago

Try this: reflect the uppermost point over the y-axis, reflect the other point over the x-axis, draw a straight line connecting them, and then reflect the points to their original positions after computing the distance.. It's a small difference but hopefully it makes the solution more obvious/intuitive!

Alain Chau - 4 years, 9 months ago

I got a slightly different answer by using a method that follows the suggestion of Mr. Reddy below. That answer is close to the one that alleged to be the correct one, but not the same. I got the answer right because I assumed my slight error was due to rounding. Now that I read Mr. Ellinor's rather vague discussion of his reasons for this answer, I think I may be right. My answer is 13.71, which is approximately the square root of 188. Draw a straight line from (6,3) to some point (x,0). Do the same from (x,0) to (y,0), then do the same from (y,0) to (2,8). The partial perimeter is related to the sum of these squared distances. Minimize this squared distance, you minimize the partial perimeter. It is a quadratic function in x and y. Minimize by taking partial derivatives wrt x and y, set each equal to zero, and you find that x=3 and y=4. So, the point goes from (6,3) to (3,0) to (0,4) to (2,8). Calculate the distances in that order and add and you get the square root of 18, 5 and the square root of 20 or 13.715, which I believe should be the correct answer.

David Dorenfeld - 3 years, 12 months ago

Error in last post. Obviously, the point on the y axis will be (0,y), not (y,0).

David Dorenfeld - 3 years, 12 months ago

David Dorenfeld
Jun 19, 2017

We want to go from (6 3) to (x 0) to (0 y) to (2 8). The squared distance from (6 3) to (x 0) is d1^2 = (6-x)^2 +9. From (x 0) to (0 y) is d2^2 = x^ + y^2. From (0 y) to (2 8) d3^2 = 4+ (y-8)^2. Define the sum of the three squared distances P(x, y) = d1^2 +d2^2 + d3^2. Minimize P with respect to x and y by taking partial derivatives and setting equal to zero. The second partials are positive and the cross partials zero, indicating a minimum. Solve the two equations to find that x=3 and y =4. Hence the path is (6 3) (3 0) (0 4) (2 8). Calculate the distances between successive points: 18^0.5, 5, and 20^0.5. The sum is 13.71, which is about equal to 188^0.5, somewhat more than the official answer. However, I believe this is a more sensible answer.

I have to admit I am wrong. The correct answer is indeed the square root of 185. Hard to see where I went wrong though, except perhaps in the assumption that minimizing the sum of squared distances meant minimizing the sum of distances. Sorry for the confusion.

David Dorenfeld - 3 years, 12 months ago

Mr. Ellinor's solution is correct, but it is very far from intuitive. Recognizing that the shortest distance would be the path travelled by light reflected off the two surfaces, you could say that the four points (6,3) (x,0) (0,y) and (2,8) should be such that the angles at each reflection should be equal. That means their tangents should be equal. Hence 3/(6-x) = y/x and x/y = 2/8-y. Solve and you get two equations 2y=8x-xy and 3x = 6y-xy. Solve and you get y = 11/8 *x. Substitute and solve and you get x=3.8 and y=5.22. The distance between (6,3) and (2,8) along the path through (3.8 0) and (0 5.22) is 13.6 which is the square root of 185. However, the solution follows the actual path followed; it uses the physical principle that was the basis of the reflection analysis of Mr. Ellinor and most importantly it gives explicitly the points on the x and y axes at which the reflection must occur

David Dorenfeld - 3 years, 12 months ago

Double Tag

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