Difference between revisions of "2023 AMC 12A Problems/Problem 9"
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− | Let <math>a</math> be the length of the shorter leg and <math>b</math> be the longer leg. By the Pythagorean theorem, we can derive that <math>a^2+b^2=2</math>. Using area we can also derive that <math>2ab=1</math>. <math>(a+b)^2=3</math> as given in the diagram, we can find that <math>(a-b)^2=1</math> because <math>4ab=2</math>. This means that <math>b+a=\sqrt{3}</math> and <math>b-a=1</math>. Adding the equations gives <math>b=\frac{\sqrt{3}+1}{2}</math> and when <math>b</math> is plugged in <math>a = \frac{\sqrt{3}-1}{2}</math>. Rationalizing the denominators gives us <math>2-\sqrt{3}</math> | + | Let <math>a</math> be the length of the shorter leg and <math>b</math> be the longer leg. By the Pythagorean theorem, we can derive that <math>a^2+b^2=2</math>. Using area we can also derive that <math>2ab=1</math>. <math>(a+b)^2=3</math> as given in the diagram, we can find that <math>(a-b)^2=1</math> because <math>4ab=2</math>. This means that <math>b+a=\sqrt{3}</math> and <math>b-a=1</math>. Adding the equations gives <math>b=\frac{\sqrt{3}+1}{2}</math> and when <math>b</math> is plugged in <math>a = \frac{\sqrt{3}-1}{2}</math>. Rationalizing the denominators gives us <math>\boxed{\textbf{(C) } 2-\sqrt{3}}</math>. |
~alphastar company | ~alphastar company |
Revision as of 22:17, 3 December 2023
- The following problem is from both the 2023 AMC 10A #11 and 2023 AMC 12A #9, so both problems redirect to this page.
Contents
[hide]Problem
A square of area is inscribed in a square of area
, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
Solution
Solution 1 (Manipulation)
Let be the length of the shorter leg and
be the longer leg. By the Pythagorean theorem, we can derive that
. Using area we can also derive that
.
as given in the diagram, we can find that
because
. This means that
and
. Adding the equations gives
and when
is plugged in
. Rationalizing the denominators gives us
.
~alphastar company
Solution 2 (Area)
Looking at the diagram, we know that the square inscribed in the square with area has area
. Thus, the area outside of the small square is
This area is composed of
congruent triangles, so we know that each triangle has an area of
.
From solution , the base has length
and the height
, which means that
.
We can turn this into a quadratic equation: .
By using the quadratic formula, we get .Therefore, the answer is
~ghfhgvghj10 (If I made any mistakes, feel free to make minor edits)
(Clarity & formatting edits by Technodoggo)
Solution 3
Let be the ratio of the shorter leg to the longer leg, and
be the length of longer leg. The length of the shorter leg will be
.
Because the sum of two legs is the side length of the outside square, we have , which means
. Using the Pythagorean Theorem for the shaded right triangle, we also have
. Solving both equations, we get
. Using
to substitute
in the second equation, we get
. Hence,
. By using the quadratic formula, we get
. Because
be the ratio of the shorter leg to the longer leg, it is always less than
. Therefore, the answer is
.
~sqroot
Solution 4
The side length of the bigger square is equal to , while the side length of the smaller square is
. Let
be the shorter leg and
be the longer one. Clearly,
, and
. Using Vieta's to build a quadratic, we get
Solving, we get
and
. Thus, we find
.
~vadava_lx
Solution 5
Let be the angle opposite the smaller leg. We want to find
.
The area of the triangle is which implies
or
. Therefore
Solution 6
Allow a, b to be the sides of a triangle. WLOG, suppose We want to find
. Notice that the area of a triangle is
, which results in
. Thus,
. However, the square of the hypotenuse of this triangle is
, but also
. We can write
as
, and then plug it in. We get
, so
. Applying the quadratic formula,
, or
. However, since
and
must both be solutions of the quadratic, since both equations were cyclic. Since
, then
, and
. To find
, we can simply find the square root of
. This is
, so the answer is
. - Sepehr2010
Solution 6
Note that each side length is and
Let the shorter side of our triangle be
, thus the longer leg is
. Hence, by the Pythagorean Theorem, we have
.
By the quadratic formula, we find . Hence, our answer is
~SirAppel ~ItsMeNoobieboy
Video Solution (⚡Under 4 min⚡)
~Education, the Study of Everything
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/N2lyYRMuZuk?si=99Ir3nUQxTkJDiVm
~Math-X
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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 |
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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.