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Difference between revisions of "2022 AMC 10A Problems/Problem 15"

(→‎Solution 2 (Brahmagupta Formula))
(→‎Solution 3 (Circumradius's Formula))
 
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<math>\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997</math>
 
<math>\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997</math>
  
−
==Solution 1==
+
==Diagram==
−
<b>DIAGRAM IN PROGRESS.
+
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(200);
 +
pair O, A, B, C, D;
 +
O = origin;
 +
A = (-25/2,0);
 +
C = (25/2,0);
 +
B = intersectionpoints(Circle(A,7),Circle(C,24))[0];
 +
D = intersectionpoints(Circle(A,15),Circle(C,20))[1];
 +
fill(Circle(O,25/2),yellow);
 +
fill(A--B--C--D--cycle,white);
 +
dot("$A$",A,1.5*W,linewidth(4));
 +
dot("$B$",B,1.5*dir(B),linewidth(4));
 +
dot("$C$",C,1.5*E,linewidth(4));
 +
dot("$D$",D,1.5*dir(D),linewidth(4));
 +
dot(O,linewidth(4));
 +
draw(Circle(O,25/2));
 +
draw(A--B--C--D--cycle);
 +
label("$7$",midpoint(A--B),rotate(90)*dir(midpoint(A--B)--A));
 +
label("$24$",midpoint(B--C),rotate(-90)*dir(midpoint(B--C)--B));
 +
label("$20$",midpoint(C--D),rotate(-90)*dir(midpoint(C--D)--C));
 +
label("$15$",midpoint(D--A),rotate(90)*dir(midpoint(D--A)--D));
 +
</asy>
 +
~MRENTHUSIASM
  
−
WILL BE DONE TOMORROW, WAIT FOR ME THANKS.</b>
+
==Solution 1 (Inscribed Angle Theorem)==
  
 
Opposite angles of every cyclic quadrilateral are supplementary, so <cmath>\angle B + \angle D = 180^{\circ}.</cmath>
 
Opposite angles of every cyclic quadrilateral are supplementary, so <cmath>\angle B + \angle D = 180^{\circ}.</cmath>
Line 24: Line 47:
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
−
== Solution 2 (Brahmagupta Formula)==
+
== Solution 2 (Brahmagupta‘s Formula)==
 +
 
 +
When we look at the side lengths of the quadrilateral we see <math>7</math> and <math>24,</math> which screams out <math>25</math> because of Pythagorean triplets. As a result, we can draw a line through points <math>A</math> and <math>C</math> to make a diameter of <math>25.</math> See Solution 1 for a rigorous proof.
 +
 
 +
This can also be shown using the Law of Cosines: Since <math>7^2+24^2-2\cdot7\cdot24\cdot\cos B=15^2+20^2-2\cdot15\cdot20\cdot\cos D</math> and <math>\cos B + \cos D = 0,</math> it follows that <math>\cos B = \cos D = 0.</math>
 +
 
 +
Since the diameter is <math>25,</math> we can see the area of the circle is just <math>\frac{625\pi}{4}</math> from the formula of the area of the circle with just a diameter.
 +
 
 +
Then we can use Brahmagupta Formula <math>\sqrt{(s - a)(s - b)(s - c)(s - d)}</math> where <math>a,b,c,d</math> are side lengths, and <math>s</math> is semi-perimeter to find the area of the quadrilateral.
 +
 
 +
You can find the answer to this quickly by noting that <math>\sqrt{(33-7)(33-24)(33-20)(33-15)} = \sqrt{(26)(9)(13)(18)} = \sqrt{2^2 \cdot 3^4 \cdot 13^2} = 2 \cdot 9 \cdot 13 = 234</math>. So now the area of the region we are trying to find is <math>\frac{625\pi}{4} - 234 = \frac{625\pi-936}{4}.</math>
  
−
When we look at the side lengths of the quadrilateral we see 7 and 24, which screams out 25 because of Pythagorean triplets. As a result, we can draw a line through points <math>A</math> and <math>C</math> to make a diameter of <math>25</math>. Since the diameter is 25, we can see the area of the circle is just <math>\frac{625\pi}{4}</math> from the formula of the area of the circle with just a diameter. Then we can use Brahmagupta Formula,  <math>\sqrt{(s - a)(s - b)(s - c)(s - c)}</math> where a,b,c,d are side lengths, and s is semi-perimeter to find the area of the quadrilateral. If we just plug the values in, we get <math>\sqrt{54756}</math> which is <math>234</math>. So now the area of the region we are trying to find is <math>\frac{625\pi}{4} - 234</math> or it can be written as <math>\frac{625\pi-936}{4}</math>
 
 
Therefore, the answer is <math>a+b+c=\boxed{\textbf{(D) } 1565}.</math>
 
Therefore, the answer is <math>a+b+c=\boxed{\textbf{(D) } 1565}.</math>
  
−
~Gdking
+
~Gdking ~Oinava ~[https://artofproblemsolving.com/wiki/index.php/User:South South]
 +
 
 +
== Solution 3 (Circumradius's Formula) ==
 +
 
 +
We can guess that this quadrilateral is actually made of two right triangles: <math>\triangle CDA</math> has a <math>3 \text{-} 4 \text{-} 5</math> ratio in the side lengths, and <math>\triangle ABC</math> is a <math>7 \text{-} 24 \text{-} 25</math> triangle.
 +
(See Solution 1 for a proof.)
 +
 
 +
Next, we can choose one of these triangles and use the circumradius formula to find the radius. Let's choose the <math>15-20-25</math> triangle. The area of the triangle is equal to the product of the side lengths divided by <math>4</math> times the circumradius. Therefore, <math>150 = \frac{15\cdot20\cdot25}{4r}</math>. Solving this simple algebraic equation gives us <math>r = \frac{25}{2}</math>.
 +
 
 +
Plugging in the values, we have <math>\frac{25}{2}^2\cdot\pi - \left(\frac{15\cdot20}{2}+\frac{7\cdot24}{2}\right) = \frac{625\cdot\pi}{4} - 234</math>. Rewriting this gives us <math>\frac{625\pi-936}{4}</math>.
 +
 
 +
Therefore, adding these values gets us <math>\boxed{\textbf{(D) } 1565}.</math>
 +
 
 +
~[[OrenSH|orenbad]]
 +
 
 +
==Solution 4 (Complete Bash mix of Brahmagupta's and some other random formulas that you probably don't know)==
 +
 
 +
To start, we can apply a couple formulas to find the circumradius and the area of the cyclic quadrilateral, and then just subtract the area of the quadrilateral from the area of the circle, and that is our final answer. This formula is really tedious with many calculations so don't do this in a contest. To find the circumradius we have: <cmath>\text{Circumradius} = \frac{1}{4} \sqrt{\frac{(ab + cd)(ac + bd)(ad + bc)}{(s-a)(s-b)(s-c)(s-d)}}</cmath>
 +
If we let <math>a = 7, b = 24, c = 20, \text{and}, d = 15</math>, that means that the semi-perimeter is just <cmath>\dfrac{7+24+20+15}{2} = 33</cmath>. Note that <math>s</math> stands for the semi-perimeter. The calculations will take very long so Im just going to skip to the answer of <math>\dfrac{25}{2}</math> as our circumradius. This means that the area of the circle is <math>\dfrac{625}{4}\pi</math>. Now applying Brahmagupta's formula to find the area of a cyclic quadrilateral which is just: <cmath>\sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath>. Again we will use the same representations as these letters from before, so <math>a = 7, b = 24, c = 20, \text{and}, d = 15</math>. Plugging in the numbers we have that the area is <math>234</math>. To express our answer in the form that was asked at the top we have: <cmath>\dfrac{625\pi-936}{4}</cmath>, so our final answer is <cmath>625+936+4 = \boxed{\textbf{(D) } 1565}.</cmath>
 +
 
 +
-jb2015007
 +
 
 +
==Video Solution 1==
 +
https://youtu.be/ZHuInvG82PY
 +
 
 +
~Education, the Study of Everything
 +
 
 +
== Video Solution 2==
 +
 
 +
https://youtu.be/Ov9AA7veKKk
 +
 
 +
== Video Solution 3==
 +
 
 +
https://youtu.be/x3DrtvR3sQ8
 +
==Video Solution by TheBeautyofMath==
 +
https://youtu.be/0kkc4-y8TkU?t=1632
  
 +
~IceMatrix
 
== See Also ==
 
== See Also ==
  
 
{{AMC10 box|year=2022|ab=A|num-b=14|num-a=16}}
 
{{AMC10 box|year=2022|ab=A|num-b=14|num-a=16}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 11:11, 13 January 2025

Problem

Quadrilateral $ABCD$ with side lengths $AB=7, BC=24, CD=20, DA=15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi-b}{c},$ where $a,b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c?$

$\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997$

Diagram

[asy] /* Made by MRENTHUSIASM */ size(200); pair O, A, B, C, D; O = origin; A = (-25/2,0); C = (25/2,0); B = intersectionpoints(Circle(A,7),Circle(C,24))[0]; D = intersectionpoints(Circle(A,15),Circle(C,20))[1]; fill(Circle(O,25/2),yellow); fill(A--B--C--D--cycle,white); dot("$A$",A,1.5*W,linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*E,linewidth(4)); dot("$D$",D,1.5*dir(D),linewidth(4)); dot(O,linewidth(4)); draw(Circle(O,25/2)); draw(A--B--C--D--cycle); label("$7$",midpoint(A--B),rotate(90)*dir(midpoint(A--B)--A)); label("$24$",midpoint(B--C),rotate(-90)*dir(midpoint(B--C)--B)); label("$20$",midpoint(C--D),rotate(-90)*dir(midpoint(C--D)--C)); label("$15$",midpoint(D--A),rotate(90)*dir(midpoint(D--A)--D)); [/asy] ~MRENTHUSIASM

Solution 1 (Inscribed Angle Theorem)

Opposite angles of every cyclic quadrilateral are supplementary, so \[\angle B + \angle D = 180^{\circ}.\] We claim that $AC=25.$ We can prove it by contradiction:

  • If $AC<25,$ then $\angle B$ and $\angle D$ are both acute angles. This arrives at a contradiction.
  • If $AC>25,$ then $\angle B$ and $\angle D$ are both obtuse angles. This arrives at a contradiction.

By the Inscribed Angle Theorem, we conclude that $\overline{AC}$ is the diameter of the circle. So, the radius of the circle is $r=\frac{AC}{2}=\frac{25}{2}.$

The area of the requested region is \[\pi r^2 - \frac12\cdot AB\cdot BC - \frac12\cdot AD\cdot DC = \frac{625\pi}{4}-\frac{168}{2}-\frac{300}{2}=\frac{625\pi-936}{4}.\] Therefore, the answer is $a+b+c=\boxed{\textbf{(D) } 1565}.$

~MRENTHUSIASM

Solution 2 (Brahmagupta‘s Formula)

When we look at the side lengths of the quadrilateral we see $7$ and $24,$ which screams out $25$ because of Pythagorean triplets. As a result, we can draw a line through points $A$ and $C$ to make a diameter of $25.$ See Solution 1 for a rigorous proof.

This can also be shown using the Law of Cosines: Since $7^2+24^2-2\cdot7\cdot24\cdot\cos B=15^2+20^2-2\cdot15\cdot20\cdot\cos D$ and $\cos B + \cos D = 0,$ it follows that $\cos B = \cos D = 0.$

Since the diameter is $25,$ we can see the area of the circle is just $\frac{625\pi}{4}$ from the formula of the area of the circle with just a diameter.

Then we can use Brahmagupta Formula $\sqrt{(s - a)(s - b)(s - c)(s - d)}$ where $a,b,c,d$ are side lengths, and $s$ is semi-perimeter to find the area of the quadrilateral.

You can find the answer to this quickly by noting that $\sqrt{(33-7)(33-24)(33-20)(33-15)} = \sqrt{(26)(9)(13)(18)} = \sqrt{2^2 \cdot 3^4 \cdot 13^2} = 2 \cdot 9 \cdot 13 = 234$. So now the area of the region we are trying to find is $\frac{625\pi}{4} - 234 = \frac{625\pi-936}{4}.$

Therefore, the answer is $a+b+c=\boxed{\textbf{(D) } 1565}.$

~Gdking ~Oinava ~South

Solution 3 (Circumradius's Formula)

We can guess that this quadrilateral is actually made of two right triangles: $\triangle CDA$ has a $3 \text{-} 4 \text{-} 5$ ratio in the side lengths, and $\triangle ABC$ is a $7 \text{-} 24 \text{-} 25$ triangle. (See Solution 1 for a proof.)

Next, we can choose one of these triangles and use the circumradius formula to find the radius. Let's choose the $15-20-25$ triangle. The area of the triangle is equal to the product of the side lengths divided by $4$ times the circumradius. Therefore, $150 = \frac{15\cdot20\cdot25}{4r}$. Solving this simple algebraic equation gives us $r = \frac{25}{2}$.

Plugging in the values, we have $\frac{25}{2}^2\cdot\pi - \left(\frac{15\cdot20}{2}+\frac{7\cdot24}{2}\right) = \frac{625\cdot\pi}{4} - 234$. Rewriting this gives us $\frac{625\pi-936}{4}$.

Therefore, adding these values gets us $\boxed{\textbf{(D) } 1565}.$

~orenbad

Solution 4 (Complete Bash mix of Brahmagupta's and some other random formulas that you probably don't know)

To start, we can apply a couple formulas to find the circumradius and the area of the cyclic quadrilateral, and then just subtract the area of the quadrilateral from the area of the circle, and that is our final answer. This formula is really tedious with many calculations so don't do this in a contest. To find the circumradius we have: \[\text{Circumradius} = \frac{1}{4} \sqrt{\frac{(ab + cd)(ac + bd)(ad + bc)}{(s-a)(s-b)(s-c)(s-d)}}\] If we let $a = 7, b = 24, c = 20, \text{and}, d = 15$, that means that the semi-perimeter is just \[\dfrac{7+24+20+15}{2} = 33\]. Note that $s$ stands for the semi-perimeter. The calculations will take very long so Im just going to skip to the answer of $\dfrac{25}{2}$ as our circumradius. This means that the area of the circle is $\dfrac{625}{4}\pi$. Now applying Brahmagupta's formula to find the area of a cyclic quadrilateral which is just: \[\sqrt{(s-a)(s-b)(s-c)(s-d)}\]. Again we will use the same representations as these letters from before, so $a = 7, b = 24, c = 20, \text{and}, d = 15$. Plugging in the numbers we have that the area is $234$. To express our answer in the form that was asked at the top we have: \[\dfrac{625\pi-936}{4}\], so our final answer is \[625+936+4 = \boxed{\textbf{(D) } 1565}.\]

-jb2015007

Video Solution 1

https://youtu.be/ZHuInvG82PY

~Education, the Study of Everything

Video Solution 2

https://youtu.be/Ov9AA7veKKk

Video Solution 3

https://youtu.be/x3DrtvR3sQ8

Video Solution by TheBeautyofMath

https://youtu.be/0kkc4-y8TkU?t=1632

~IceMatrix

See Also

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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

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