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Difference between revisions of "2022 SSMO Team Round Problems/Problem 5"

(Created page with "==Problem== Consider the following rectangle <math>ABCD</math> where <math>BC=8.</math> If<cmath>CD=CT_2, 4T_2P_2=2DP_1=AC, [ADP_1C]=33, \text{ and }[ABP_2C]=34,</cmath>find t...")
 
 
Line 1: Line 1:
 
==Problem==
 
==Problem==
−
Consider the following rectangle <math>ABCD</math> where <math>BC=8.</math> If<cmath>CD=CT_2, 4T_2P_2=2DP_1=AC, [ADP_1C]=33, \text{ and }[ABP_2C]=34,</cmath>find the value of <math>[P_1CP_2A].</math> (Note that <math>[ABC]</math> is the area of <math>ABC</math>.)
+
Consider the following rectangle <math>ABCD</math> where <math>BC=8.</math> If <cmath>CD=CT_2, 4T_2P_2=2DP_1=AC, [ADP_1C]=33, \text{ and }[ABP_2C]=34,</cmath> find the value of <math>[P_1CP_2A].</math> (Note that <math>[ABC]</math> is the area of <math>ABC</math>.)
−
 
+
<center>
−
[asy] unitsize(0.25cm); dot((0,0)); dot((12,0)); dot((12,16)); dot((0,16)); draw((0,0)--(12,0)--(12,16)--(0,16)--(0,0)--cycle); dot((8,22)); dot((17,4)); dot((8,16)); dot((12,4)); label("<math>A</math>", (0,0), SW); label("<math>B</math>", (12,0), SE); label("<math>C</math>", (12,16), NE); label("<math>D</math>", (0,16), NW); label("<math>P_1</math>", (8,22), NE); label("<math>P_2</math>", (17,4), SE); label("<math>T_1</math>", (8,16), SE); label("<math>T_2</math>", (12,4), SW); draw((8,22)--(8,16),dashed+linewidth(.5)); draw((12,4)--(17,4),dashed+linewidth(.5)); [/asy]
+
<asy>
 +
unitsize(0.25cm);
 +
dot((0,0));
 +
dot((12,0));
 +
dot((12,16));
 +
dot((0,16));
 +
draw((0,0)--(12,0)--(12,16)--(0,16)--(0,0)--cycle);
 +
dot((8,22));
 +
dot((17,4));
 +
dot((8,16));
 +
dot((12,4));
 +
label("$A$", (0,0), SW);
 +
label("$B$", (12,0), SE);
 +
label("$C$", (12,16), NE);
 +
label("$D$", (0,16), NW);
 +
label("$P_1$", (8,22), NE);
 +
label("$P_2$", (17,4), SE);
 +
label("$T_1$", (8,16), SE);
 +
label("$T_2$", (12,4), SW);
 +
draw((8,22)--(8,16),dashed+linewidth(.5));
 +
draw((12,4)--(17,4),dashed+linewidth(.5));
 +
</asy>
 +
</center>
  
 
==Solution==
 
==Solution==

Latest revision as of 12:09, 14 December 2023

Problem

Consider the following rectangle $ABCD$ where $BC=8.$ If \[CD=CT_2, 4T_2P_2=2DP_1=AC, [ADP_1C]=33, \text{ and }[ABP_2C]=34,\] find the value of $[P_1CP_2A].$ (Note that $[ABC]$ is the area of $ABC$.)

[asy] unitsize(0.25cm); dot((0,0)); dot((12,0)); dot((12,16)); dot((0,16)); draw((0,0)--(12,0)--(12,16)--(0,16)--(0,0)--cycle); dot((8,22)); dot((17,4)); dot((8,16)); dot((12,4)); label("$A$", (0,0), SW); label("$B$", (12,0), SE); label("$C$", (12,16), NE); label("$D$", (0,16), NW); label("$P_1$", (8,22), NE); label("$P_2$", (17,4), SE); label("$T_1$", (8,16), SE); label("$T_2$", (12,4), SW); draw((8,22)--(8,16),dashed+linewidth(.5)); draw((12,4)--(17,4),dashed+linewidth(.5)); [/asy]

Solution

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