# Distance to the corner of a rectangle

## Homework Statement

This question is taken from 2011 Malaysian Mathematical Olympiad.
Mary is standing in a rectangular garden. Her distance to the four corners of the garden are 6 m, 7 m, 9 m and d m, respectively, where d is an integer. Find d.

## Homework Equations

Triangle inequality. a + b < c, a + c < b, b + c < a, where a, b, and c are the lengths of the three sides of the triangle.

## The Attempt at a Solution

I tried to denote the length of the rectangular garden as a and b, respectively, then from the four triangles formed, I formed some inequality and try to see if the value of d is bounded, but it yields nothing. I have also tried to solve for d by using the concept of area. Also, I tried using the law of cosine and the Pythagorean theorem. But still, I can't find the value for d.

Any other ideas how to approach this problem? Thanks. :)

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eumyang
Homework Helper
Pythagorean Theorem is the way to go. Draw two perpendicular lines "through" Mary. You'll have four right triangles, with 6, 7, 9 and d being the hypotenuses. Use the Pythagorean Theorem four times, and through some manipulation, you'll be able to find d.

Pythagorean Theorem is the way to go. Draw two perpendicular lines "through" Mary. You'll have four right triangles, with 6, 7, 9 and d being the hypotenuses. Use the Pythagorean Theorem four times, and through some manipulation, you'll be able to find d.
Thanks for the clue. Now I have a clearer direction. I get d = √94, not an integer though, but still, at least I can get the value of d. :)

eumyang
Homework Helper
Thanks for the clue. Now I have a clearer direction. I get d = √94, not an integer though, but still, at least I can get the value of d. :)
I didn't get that answer. Can you double-check?

EDIT: I think I know why our answers differ. It depends on how you label the four distances from Mary to the corners. I took "6, 7, 9 and d, respectively" to mean that you label the line segments clockwise in that fashion. It looks like you labeled them as "6, 9, d and 7," going clockwise, or something similar. Are you looking at a diagram?

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$v_{1}^{2}+h_{1}^{2}=6^2=36$
$v_{1}^{2}+h_{2}^{2}=7^2=49$
$v_{2}^{2}+h_{1}^{2}=9^2=81$
$v_{2}^{2}+h_{2}^{2}=d^2$

$d^2=v{2}^{2}+h_{2}^{2}=130-36=94$

EDIT: I think I know why our answers differ. It depends on how you label the four distances from Mary to the corners. I took "6, 7, 9 and d, respectively" to mean that you label the line segments clockwise in that fashion. It looks like you labeled them as "6, 9, d and 7," going clockwise, or something similar. Are you looking at a diagram?
Yup. I should have used the "6, 7, 9, d" clockwise. I recount, and get √68. :)

EDIT: No diagram was given for the question.

Since the question says d is an integer, I try to use other types of 'combination', and when I tried with '6, 9, 7, d' going clockwise, I get d = 2. Yay! Thanks again, eumyang, for the help :)

eumyang
Homework Helper