Difference between revisions of "2017 AMC 10A Problems/Problem 21"
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− | A square with side length <math>x</math> is inscribed in a right triangle with sides of length <math>3</math>, <math>4</math>, and <math>5</math> so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length <math>y</math> is inscribed in another right triangle with sides of length <math>3</math>, <math>4</math>, and <math>5</math> so that one side of the square lies on the hypotenuse of the triangle. What is <math>\ | + | ==Problem== |
+ | A square with side length <math>x</math> is inscribed in a right triangle with sides of length <math>3</math>, <math>4</math>, and <math>5</math> so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length <math>y</math> is inscribed in another right triangle with sides of length <math>3</math>, <math>4</math>, and <math>5</math> so that one side of the square lies on the hypotenuse of the triangle. What is <math>\dfrac{x}{y}</math>? | ||
<math>\textbf{(A) } \dfrac{12}{13} \qquad \textbf{(B) } \dfrac{35}{37} \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \dfrac{37}{35} \qquad \textbf{(E) } \dfrac{13}{12}</math> | <math>\textbf{(A) } \dfrac{12}{13} \qquad \textbf{(B) } \dfrac{35}{37} \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \dfrac{37}{35} \qquad \textbf{(E) } \dfrac{13}{12}</math> | ||
− | ==Solution== | + | ==Solution 1== |
Analyze the first right triangle. | Analyze the first right triangle. | ||
Line 60: | Line 61: | ||
</asy> | </asy> | ||
− | + | Similarly, <math>\triangle A'B'C'</math> and <math>\triangle RB'Q</math> are similar, so <math>RB' = \frac{4}{3}y</math>, and <math>C'S = \frac{3}{4}y</math>. Thus, <math>C'B' = C'S + SR + RB' = \frac{4}{3}y + y + \frac{3}{4}y = 5</math>. Solving for <math>y</math>, we get <math>y = \frac{60}{37}</math>. Thus, <math>\frac{x}{y} = \boxed{\textbf{(D)}\:\frac{37}{35}}</math>. | |
+ | |||
+ | ==Solution 2 (Alternate solution in finding x)== | ||
+ | Set the right-angle vertex of the triangle as <math>(0,0)</math>. Notice that the hypotenuse of the triangle, as depicted in solution one, can be described by <math>y = 3 - \frac{3}{4}x</math>, while <math>AE</math> can be describe by <math>y=x</math>. Hence, we may solve for <math>x</math> by solving <math> 3- \frac{3}{4}x = x</math>, which yields <math>\frac{12}{7}</math>. | ||
+ | |||
+ | Proceed by finding the value of y via the method described in solution 1, and we will get <math>y = \frac{60}{37}</math>. Thus, <math>\frac{x}{y} = \boxed{\textbf{(D)}\:\frac{37}{35}}</math>. | ||
+ | |||
+ | ==Note== | ||
+ | In general, if the legs were <math>a</math> and <math>b</math>, we have that | ||
+ | <cmath>x= \frac{ab}{a+b}, y=\frac{ab\sqrt{a^2+b^2}}{a^2+ab+b^2}.</cmath> | ||
+ | This can be verified by plugging in <math>a=3</math> and <math>b=4</math>. | ||
+ | ~anduran | ||
+ | |||
+ | ==Video Solution by Pi Academy== | ||
+ | |||
+ | https://youtu.be/DJ1105lcJJM?si=Z24jb7mCjzjLPdKh | ||
+ | |||
+ | ~ Pi Academy | ||
+ | |||
+ | ==Other video solutions== | ||
+ | https://youtu.be/THeq4ZiZxIA | ||
+ | -Video Solution by TheBeautyOfMath | ||
+ | |||
+ | https://youtu.be/MF2QFOInbYc | ||
+ | -Video Solution by Richard Rusczyk | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2017|ab=A|num-b=20|num-a=22}} | ||
+ | {{AMC12 box|year=2017|ab=A|num-b=18|num-a=20}} | ||
+ | {{MAA Notice}} | ||
+ | |||
+ | [[Category:Introductory Geometry Problems]] |
Latest revision as of 15:20, 11 October 2024
Contents
Problem
A square with side length is inscribed in a right triangle with sides of length , , and so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length is inscribed in another right triangle with sides of length , , and so that one side of the square lies on the hypotenuse of the triangle. What is ?
Solution 1
Analyze the first right triangle.
Note that and are similar, so . This can be written as . Solving, .
Now we analyze the second triangle.
Similarly, and are similar, so , and . Thus, . Solving for , we get . Thus, .
Solution 2 (Alternate solution in finding x)
Set the right-angle vertex of the triangle as . Notice that the hypotenuse of the triangle, as depicted in solution one, can be described by , while can be describe by . Hence, we may solve for by solving , which yields .
Proceed by finding the value of y via the method described in solution 1, and we will get . Thus, .
Note
In general, if the legs were and , we have that This can be verified by plugging in and . ~anduran
Video Solution by Pi Academy
https://youtu.be/DJ1105lcJJM?si=Z24jb7mCjzjLPdKh
~ Pi Academy
Other video solutions
https://youtu.be/THeq4ZiZxIA -Video Solution by TheBeautyOfMath
https://youtu.be/MF2QFOInbYc -Video Solution by Richard Rusczyk
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 |
2017 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.