Register to reply

Distance to the corner of a rectangle

by lkh1986
Tags: corner, distance, rectangle
Share this thread:
lkh1986
#1
Nov20-11, 07:15 AM
P: 99
1. The problem statement, all variables and given/known data

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.


2. Relevant 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.



3. 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. :)
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
eumyang
#2
Nov20-11, 07:41 AM
HW Helper
P: 1,347
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.
lkh1986
#3
Nov20-11, 07:53 AM
P: 99
Quote Quote by eumyang View Post
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
#4
Nov20-11, 08:02 AM
HW Helper
P: 1,347
Distance to the corner of a rectangle

Quote Quote by lkh1986 View Post
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?
lkh1986
#5
Nov20-11, 08:10 AM
P: 99
[itex]v_{1}^{2}+h_{1}^{2}=6^2=36[/itex]
[itex]v_{1}^{2}+h_{2}^{2}=7^2=49[/itex]
[itex]v_{2}^{2}+h_{1}^{2}=9^2=81[/itex]
[itex]v_{2}^{2}+h_{2}^{2}=d^2[/itex]

[itex]d^2=v{2}^{2}+h_{2}^{2}=130-36=94[/itex]
lkh1986
#6
Nov20-11, 08:14 AM
P: 99
Quote Quote by eumyang View Post

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.
lkh1986
#7
Nov20-11, 08:24 AM
P: 99
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
#8
Nov20-11, 08:28 AM
HW Helper
P: 1,347
I had forgotten about the "d is an integer" part when I last posted. I'm glad you got the answer.


Register to reply

Related Discussions
How to calc sides of rectangle inside a rotated rectangle General Math 5
Rectangle inscribed in another rectangle: solutions for all cases. General Math 14
Finding the dimensions of a rotated rectangle inside another rectangle. Precalculus Mathematics Homework 8
Car acceleration around a corner Introductory Physics Homework 29
Car going round corner Introductory Physics Homework 4