Difference between revisions of "2008 iTest Problems/Problem 24"
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Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground). | Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground). | ||
| − | ==Solution== | + | ==Solution 1== |
<asy> | <asy> | ||
| Line 43: | Line 43: | ||
</asy> | </asy> | ||
| − | Note that because the unknown length is the furthest | + | Note that because the unknown length is the furthest diagonal, draw the diagram as shown because it maximizes the unknown length. By the [[Pythagorean Theorem]], |
<cmath>a^2 + b^2 = 81</cmath> | <cmath>a^2 + b^2 = 81</cmath> | ||
<cmath>a^2 + d^2 = 225</cmath> | <cmath>a^2 + d^2 = 225</cmath> | ||
| Line 53: | Line 53: | ||
<cmath>c^2 + d^2 = 400</cmath> | <cmath>c^2 + d^2 = 400</cmath> | ||
Thus, the distance from the center of the staircase to the furthest corner of the castle is <math>\boxed{20}</math> inches. | Thus, the distance from the center of the staircase to the furthest corner of the castle is <math>\boxed{20}</math> inches. | ||
| + | |||
| + | ==Solution 2== | ||
| + | By the [[British Flag Theorem]], the length of the longest diagonal is <math>\sqrt{16^2+15^2-9^2}=\sqrt{400}=\boxed{20}</math> | ||
==See Also== | ==See Also== | ||
Latest revision as of 00:24, 2 November 2023
Contents
Problem
In order to earn her vacation spending money, Alexis helped her mother remove weeds from the garden. When she was done, she came into the house to put away her gardening gloves and change into clean clothes.
On her way to her room she notices Joshua with his face to the floor in the family room, looking pretty silly. "Josh, did you know you lose IQ points for sniffing the carpet?"
"Shut up. I'm not sniffing the carpet. I'm doing something."
"Sure, if sniffing the carpet counts as doing something." At this point Alexis stands over her twin brother grinning, trying to see how silly she can make him feel.
Joshua climbs to his feet and stands on his toes to make himself a half inch taller than his sister, who is ordinarily a half inch taller than Joshua. "I'm measuring something. I'm designing something."
Alexis stands on her toes too, reminding her brother that she is still taller than he. "When you're done, can you design me a dress?"
"Very funny." Joshua walks to the table and points to some drawings. "I'm designing the sand castle I want to build at the beach. Everything needs to be measured out so that I can build something awesome."
"And this requires sniffing carpet?" inquires Alexis, who is just a little intrigued by her brother's project.
"I was imagining where to put the base of a spiral staircase. Everything needs to be measured out correctly. See, the castle walls will be in the shape of a rectangle, like this room. The center of the staircase will be 
inches from one of the corners, 
inches from another, 16 inches from another, and some whole number of inches from the furthest corner." Joshua shoots Alexis a wry smile. The twins liked to challenge each other, and Alexis knew she had to find the distance from the center of the staircase to the fourth corner of the castle on her own, or face Joshua's pestering, which might last for hours or days.
Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground).
Solution 1
![[asy] draw((0,0)--(50,0)--(50,60)--(0,60)--(0,0)); dot((16,32)); draw((16,32)--(0,60)); label("9",(8,46),SW); draw((16,32)--(0,0)); label("15",(8,16),NW); draw((16,32)--(50,60)); label("16",(33,46),NW); draw((16,32)--(50,0)); draw((0,32)--(50,32),dotted); label("$a$",(8,32),S); label("$c$",(33,32),S); draw((16,0)--(16,60),dotted); label("$b$",(16,46),E); label("$d$",(16,16),E); [/asy]](http://latex.artofproblemsolving.com/5/2/3/52339ed37ec5a267965d1c554fbb005a258456bd.png)
Note that because the unknown length is the furthest diagonal, draw the diagram as shown because it maximizes the unknown length. By the Pythagorean Theorem,
![\[a^2 + b^2 = 81\]](http://latex.artofproblemsolving.com/3/e/9/3e95184c6f5c8e4b7d70d459599d1da9e8c6bd71.png)
![\[a^2 + d^2 = 225\]](http://latex.artofproblemsolving.com/4/d/3/4d379f2b9f23cb012f7b47de90b204c860e83750.png)
![\[b^2 + c^2 = 256\]](http://latex.artofproblemsolving.com/d/9/8/d986b10f38a532972fee45ed4fe35a2333f96547.png)
Adding the second and third equations results in
![\[a^2 + b^2 + c^2 + d^2 = 481\]](http://latex.artofproblemsolving.com/4/8/2/48296185b775b66deb3ae57996eb13e2c1e6f3f3.png)
By substitution,
![\[81 + c^2 + d^2 = 481\]](http://latex.artofproblemsolving.com/2/f/2/2f2de71f7bd3d3377234543a1fd61e561de752e8.png)
![\[c^2 + d^2 = 400\]](http://latex.artofproblemsolving.com/9/5/a/95a024a47541bc891e2af35bf435ba7d9cb561d0.png)
Thus, the distance from the center of the staircase to the furthest corner of the castle is 
inches.
Solution 2
By the British Flag Theorem, the length of the longest diagonal is 
See Also
| 2008 iTest (Problems) | ||
| Preceded by: Problem 23 |
Followed by: Problem 25 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 • 100 | ||
