Difference between revisions of "2022 AMC 10A Problems/Problem 10"
Eashanshetty (talk | contribs) (→Video Solution 1 (Quick and Easy)) |
|||
Line 58: | Line 58: | ||
Edits and Diagram by ~KingRavi | Edits and Diagram by ~KingRavi | ||
− | ==Video Solution 1 (Quick and Easy)== | + | ==Video Solution 1 (Simple) |
+ | https://www.youtube.com/watch?v=joVRkVp7Qvc | ||
+ | ~AWhiz | ||
+ | |||
+ | ==Video Solution 2 (Quick and Easy)== | ||
https://youtu.be/BIy0Koe4D4s | https://youtu.be/BIy0Koe4D4s | ||
Revision as of 11:51, 19 November 2022
Contents
Problem
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be centimeters, as shown below. What is the area of the original index card?
Solution 1(Simple coordinates and basic algebra)
We will use coordinates here. Label the bottom left corner of the larger rectangle(without the square cut out) as and the top right as , where is the width of the rectangle and is the length. Now we have vertices , , , and as vertices of the irregular octagon created by cutting out the squares. Label and as the two closest vertices formed by the squares. The distance between the two closest vertices of the squares is thus = ( Substituting, we get
Using the fact that the diagonal of the rectangle = , we get
.
Subtracting the first equation from the second equation, we get Squaring yields Subtracting the second equation from this, we get , and thus area of the original rectangle = = =
~USAMO333
Edits and Diagram by ~KingRavi
==Video Solution 1 (Simple) https://www.youtube.com/watch?v=joVRkVp7Qvc ~AWhiz
Video Solution 2 (Quick and Easy)
~Education, the Study of Everything
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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 | ||
All AMC 10 Problems and Solutions |