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MATH / Isosceles △ABC has a right angle at C. Point P is inside △ABC, such that PA=11, PB=7, and PC=6. Legs (AC)¯ and (BC)¯ have length s=√(a+b√(2)), where a and b are positive integers. …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Isosceles △ABC\triangle{ABC} has a right angle at CC. Point PP is inside △ABC\triangle{ABC}, such that PA=11PA=11, PB=7PB=7, and PC=6PC=6. Legs AC¯\overline{AC} and BC¯\overline{BC} have length s=a+b2s=\sqrt{a+b\sqrt{2}}, where aa and bb are positive integers. What is a+ba+b? [asy] pair A,B,C,P; A=(10,0); B=(0,10); C=(0,0); P=(3,3.5); draw(A--B--C--cycle,linewidth(0.7)); draw(A--P,linewidth(0.7)); draw(B--P--C,linewidth(0.7)); label("AA",A,E); label("BB",B,N); label("CC",C,S); label("PP",P,NE); label("7",(1.5,6.75),E); label("6",(1.5, 1.75),E); label("11",(6.5,1.75),S); [/asy]
Plain-text mathematical notation (without MathML)
Isosceles △ABC has a right angle at C.  Point P is inside △ABC, such that PA=11, PB=7, and PC=6. Legs (AC)¯ and (BC)¯ have length s=√(a+b√(2)), where a and b are positive integers. What is a+b?

[asy]
pair A,B,C,P;
A=(10,0);
B=(0,10);
C=(0,0);
P=(3,3.5);
draw(A--B--C--cycle,linewidth(0.7));
draw(A--P,linewidth(0.7));
draw(B--P--C,linewidth(0.7));
label("A",A,E);
label("B",B,N);
label("C",C,S);
label("P",P,NE);
label("7",(1.5,6.75),E);
label("6",(1.5, 1.75),E);
label("11",(6.5,1.75),S);
[/asy]
Original LaTeX notation
Isosceles $\triangle{ABC}$ has a right angle at $C$.  Point $P$ is inside $\triangle{ABC}$, such that $PA=11$, $PB=7$, and $PC=6$. Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?

[asy]
pair A,B,C,P;
A=(10,0);
B=(0,10);
C=(0,0);
P=(3,3.5);
draw(A--B--C--cycle,linewidth(0.7));
draw(A--P,linewidth(0.7));
draw(B--P--C,linewidth(0.7));
label("$A$",A,E);
label("$B$",B,N);
label("$C$",C,S);
label("$P$",P,NE);
label("7",(1.5,6.75),E);
label("6",(1.5, 1.75),E);
label("11",(6.5,1.75),S);
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
875bc345-b0d0-507b-a27e-e88b3d4a2461
task_key
geometry--test--875bc345-b0d0-507b-a27e-e88b3d4a2461
task_revision_id
1
config / split
geometry / test

Official source

initial import