Problem

2021 Fall AMC 10B Problem 15

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]

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

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Category: AMC 10B
Points: 3
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