Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 Fall AMC 10B Problems/Problem 15"

(→‎Solution 3 (Analytical geometry: point-line distance formula))
Line 22: Line 22:
 
<math>\textbf{(A) }85\qquad\textbf{(B) }93\qquad\textbf{(C) }100\qquad\textbf{(D) }117\qquad\textbf{(E) }125</math>
 
<math>\textbf{(A) }85\qquad\textbf{(B) }93\qquad\textbf{(C) }100\qquad\textbf{(D) }117\qquad\textbf{(E) }125</math>
  
−
==Solution==
+
==Solution 1==
  
 
Note that <math>\triangle APB \cong \triangle BQC.</math> Then, it follows that <math>\overline{PB} \cong \overline{QC}.</math> Thus, <math>QC = PB = PR + RB = 7 + 6 = 13.</math> Define <math>x</math> to be the length of side <math>CR,</math> then <math>RQ = 13-x.</math> Because <math>\overline{BR}</math> is the altitude of the triangle, we can use the property that <math>QR \cdot RC = BR^2.</math> Substituting the given lengths, we have <cmath>(13-x) \cdot x = 36.</cmath> Solving, gives <math>x = 4</math> and <math>x = 9.</math> We eliminate the possibilty of <math>x=4</math> because <math>RC > QR.</math> Thus, the side lengnth of the square, by Pythagorean Theorem, is <cmath>\sqrt{9^2 +6^2} = \sqrt{81+36} = \sqrt{117}.</cmath> Thus, the area of the sqaure is <math>(\sqrt{117})^2 = 117.</math> Thus, the answer is <math>\boxed{(\textbf{D}.)}.</math>  
 
Note that <math>\triangle APB \cong \triangle BQC.</math> Then, it follows that <math>\overline{PB} \cong \overline{QC}.</math> Thus, <math>QC = PB = PR + RB = 7 + 6 = 13.</math> Define <math>x</math> to be the length of side <math>CR,</math> then <math>RQ = 13-x.</math> Because <math>\overline{BR}</math> is the altitude of the triangle, we can use the property that <math>QR \cdot RC = BR^2.</math> Substituting the given lengths, we have <cmath>(13-x) \cdot x = 36.</cmath> Solving, gives <math>x = 4</math> and <math>x = 9.</math> We eliminate the possibilty of <math>x=4</math> because <math>RC > QR.</math> Thus, the side lengnth of the square, by Pythagorean Theorem, is <cmath>\sqrt{9^2 +6^2} = \sqrt{81+36} = \sqrt{117}.</cmath> Thus, the area of the sqaure is <math>(\sqrt{117})^2 = 117.</math> Thus, the answer is <math>\boxed{(\textbf{D}.)}.</math>  
Line 29: Line 29:
  
 
==Solution 2==
 
==Solution 2==
 +
 +
Solution in Progress
 +
~KingRavi
 +
 +
==Solution 3==
  
 
We have that <math>\triangle CRB \sim \triangle BAP.</math> Thus, <math>\frac{\overline{CB}}{\overline{CR}} = \frac{\overline{PB}}{\overline{AB}}</math>. Now, let the side length of the square be <math>s.</math> Then, by the Pythagorean theorem, <math>CR = \sqrt{x^2-36}.</math> Plugging all of this information in, we get <cmath>\frac{s}{\sqrt{s^2-36}} = \frac{13}{s}.</cmath> Simplifying gives <cmath>s^2=13\sqrt{s^2-36},</cmath> Squaring both sides gives <cmath>s^4 = 169s^2- 169\cdot 36 \implies s^4-169s^2 + 169\cdot 36 = 0.</cmath> We now set <math>s^2=t,</math> and get the equation <math>t^2-169t + 169\cdot 36 = 0.</math> From here, notice we want to solve for <math>t</math>, as it is precisely <math>s^2,</math> or the area of the square. So we use the [[Quadratic formula]], and though it may seem bashy, we hope for a nice cancellation of terms. <cmath>t = \frac{169\pm\sqrt{169^2-4\cdot 36 \cdot 169}}{2}.</cmath> It seems scary, but factoring <math>169</math> from the square root gives us <cmath>t = \frac{169\pm \sqrt{169 \cdot (169-144)}}{2} = \frac{169 \pm \sqrt{169 \cdot 25}}{2} = \frac{169 \pm 13\cdot 5}{2} = \frac{169\pm 65}{2},</cmath> giving us the solutions <cmath>t=52, 117.</cmath> We instantly see that <math>t=52</math> is way too small to be an area of this square (<math>52</math> isn't even an answer choice, so you can skip this step if out of time) because then the side length would be <math>2\sqrt{13}</math> and then, even the largest line you can draw inside the square (the diagonal)  is <math>2\sqrt{26},</math> which is less than <math>13</math> (line <math>PB</math>) And thus, <math>t</math> must be <math>117</math>, and our answer is <math>\boxed{\textbf{(D)}}.</math> <math>\blacksquare</math>
 
We have that <math>\triangle CRB \sim \triangle BAP.</math> Thus, <math>\frac{\overline{CB}}{\overline{CR}} = \frac{\overline{PB}}{\overline{AB}}</math>. Now, let the side length of the square be <math>s.</math> Then, by the Pythagorean theorem, <math>CR = \sqrt{x^2-36}.</math> Plugging all of this information in, we get <cmath>\frac{s}{\sqrt{s^2-36}} = \frac{13}{s}.</cmath> Simplifying gives <cmath>s^2=13\sqrt{s^2-36},</cmath> Squaring both sides gives <cmath>s^4 = 169s^2- 169\cdot 36 \implies s^4-169s^2 + 169\cdot 36 = 0.</cmath> We now set <math>s^2=t,</math> and get the equation <math>t^2-169t + 169\cdot 36 = 0.</math> From here, notice we want to solve for <math>t</math>, as it is precisely <math>s^2,</math> or the area of the square. So we use the [[Quadratic formula]], and though it may seem bashy, we hope for a nice cancellation of terms. <cmath>t = \frac{169\pm\sqrt{169^2-4\cdot 36 \cdot 169}}{2}.</cmath> It seems scary, but factoring <math>169</math> from the square root gives us <cmath>t = \frac{169\pm \sqrt{169 \cdot (169-144)}}{2} = \frac{169 \pm \sqrt{169 \cdot 25}}{2} = \frac{169 \pm 13\cdot 5}{2} = \frac{169\pm 65}{2},</cmath> giving us the solutions <cmath>t=52, 117.</cmath> We instantly see that <math>t=52</math> is way too small to be an area of this square (<math>52</math> isn't even an answer choice, so you can skip this step if out of time) because then the side length would be <math>2\sqrt{13}</math> and then, even the largest line you can draw inside the square (the diagonal)  is <math>2\sqrt{26},</math> which is less than <math>13</math> (line <math>PB</math>) And thus, <math>t</math> must be <math>117</math>, and our answer is <math>\boxed{\textbf{(D)}}.</math> <math>\blacksquare</math>
Line 35: Line 40:
  
  
−
==Solution 3 (Point-line distance formula)==
+
==Solution 4 (Point-line distance formula)==
  
 
[[Image:2021FallAMC10B15.png|center|480px]]
 
[[Image:2021FallAMC10B15.png|center|480px]]

Revision as of 18:34, 25 November 2021

Problem

In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square?

[asy] size(170); defaultpen(linewidth(0.6)+fontsize(10)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); dot("$Q$",Q,S); dot("$P$",P,W); dot("$R$",R,1.3*S); label("$7$",(P+R)/2,NE); label("$6$",(R+B)/2,NE); [/asy]

$\textbf{(A) }85\qquad\textbf{(B) }93\qquad\textbf{(C) }100\qquad\textbf{(D) }117\qquad\textbf{(E) }125$

Solution 1

Note that $\triangle APB \cong \triangle BQC.$ Then, it follows that $\overline{PB} \cong \overline{QC}.$ Thus, $QC = PB = PR + RB = 7 + 6 = 13.$ Define $x$ to be the length of side $CR,$ then $RQ = 13-x.$ Because $\overline{BR}$ is the altitude of the triangle, we can use the property that $QR \cdot RC = BR^2.$ Substituting the given lengths, we have \[(13-x) \cdot x = 36.\] Solving, gives $x = 4$ and $x = 9.$ We eliminate the possibilty of $x=4$ because $RC > QR.$ Thus, the side lengnth of the square, by Pythagorean Theorem, is \[\sqrt{9^2 +6^2} = \sqrt{81+36} = \sqrt{117}.\] Thus, the area of the sqaure is $(\sqrt{117})^2 = 117.$ Thus, the answer is $\boxed{(\textbf{D}.)}.$

~NH14

Solution 2

Solution in Progress ~KingRavi

Solution 3

We have that $\triangle CRB \sim \triangle BAP.$ Thus, $\frac{\overline{CB}}{\overline{CR}} = \frac{\overline{PB}}{\overline{AB}}$. Now, let the side length of the square be $s.$ Then, by the Pythagorean theorem, $CR = \sqrt{x^2-36}.$ Plugging all of this information in, we get \[\frac{s}{\sqrt{s^2-36}} = \frac{13}{s}.\] Simplifying gives \[s^2=13\sqrt{s^2-36},\] Squaring both sides gives \[s^4 = 169s^2- 169\cdot 36 \implies s^4-169s^2 + 169\cdot 36 = 0.\] We now set $s^2=t,$ and get the equation $t^2-169t + 169\cdot 36 = 0.$ From here, notice we want to solve for $t$, as it is precisely $s^2,$ or the area of the square. So we use the Quadratic formula, and though it may seem bashy, we hope for a nice cancellation of terms. \[t = \frac{169\pm\sqrt{169^2-4\cdot 36 \cdot 169}}{2}.\] It seems scary, but factoring $169$ from the square root gives us \[t = \frac{169\pm \sqrt{169 \cdot (169-144)}}{2} = \frac{169 \pm \sqrt{169 \cdot 25}}{2} = \frac{169 \pm 13\cdot 5}{2} = \frac{169\pm 65}{2},\] giving us the solutions \[t=52, 117.\] We instantly see that $t=52$ is way too small to be an area of this square ($52$ isn't even an answer choice, so you can skip this step if out of time) because then the side length would be $2\sqrt{13}$ and then, even the largest line you can draw inside the square (the diagonal) is $2\sqrt{26},$ which is less than $13$ (line $PB$) And thus, $t$ must be $117$, and our answer is $\boxed{\textbf{(D)}}.$ $\blacksquare$

~wamofan


Solution 4 (Point-line distance formula)

2021FallAMC10B15.png

Denote $a = RC$. Now tilt your head to the right and view $R, \overrightarrow{RB}$ and $\overrightarrow{RC}$ as the origin, $x$-axis and $y$-axis, respectively. In particular, we have points $B(6,0), C(0,a), P(-7,0)$. Note that side length of the square $ABCD$ is $BC = \sqrt{a^2 + 36}$. Also equation of line $BC$ is \[\underbrace{\frac{x}{6} + \frac{y}{a} = 1}_{\text{intercepts form}} \quad \implies \quad ax + 6y - 6a = 0.\] Because the distance from $P(-7,0)$ to line $\color[rgb]{0,0.4,0.65}BC: ax + 6y - 6a = 0$ is also the side length $\sqrt{a^2 + 36}$, we can apply the point-line distance formula to get \[\frac{|a\cdot(-7) + 6 \cdot 0 - 6a|}{\sqrt{a^2 + 36}} = {\sqrt{a^2 + 36}}\] which reduces to $|13a| = a^2 + 36$. Since $a$ is positive, the last equations factors as $a^2 - 13a + 36 = (a-4)(a-9) = 0$. Now judging from the figure, we learn that $a > RB = 6$. So $a = 9$. Therefore, the area of the square $ABCD$ is $BC^2 = RC^2 + RB^2 = a^2 + 6^2 = \mathbf{117}$. Choose $\boxed{(\textbf{D})\;\mathbf{117}}$. $\blacksquare$

~VensL.

See Also

2021 Fall AMC 10B (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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_Fall_AMC_10B_Problems/Problem_15&oldid=166352"