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MATH / In △ABC, AB=13, AC=5 and BC=12. Points M and N lie on (AC)¯ and (BC)¯, respectively, with CM=CN=4. Points J and K are on (AB)¯ so that (MJ)¯ and (NK)¯ are perpendicular to (AB)¯. W…

Problem

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

problem

In △ABC\triangle ABC, AB=13AB = 13, AC=5AC=5 and BC=12BC=12. Points MM and NN lie on AC¯\overline{AC} and BC¯\overline{BC}, respectively, with CM=CN=4CM = CN = 4. Points JJ and KK are on AB¯\overline{AB} so that MJ¯\overline{MJ} and NK¯\overline{NK} are perpendicular to AB¯\overline{AB}. What is the area of pentagon CMJKNCMJKN? Express your answer as a common fraction. [asy] pair A,B,C,N,K,M,J; A=(0,5); B=(12,0); C=(0,0); N=(5,0); M=(0,4); J=(0.35,4.84); K=(6,2.4); draw(A--B--C--cycle,linewidth(0.7)); draw(N--K,linewidth(0.7)); draw(M--J,linewidth(0.7)); label("AA",A,NW); label("MM",M,W); label("CC",C,SW); label("NN",N,S); label("BB",B,SE); label("KK",K,NE); label("JJ",J,NE); [/asy]
Plain-text mathematical notation (without MathML)
In △ABC, AB=13, AC=5 and BC=12.  Points M and N lie on (AC)¯ and (BC)¯, respectively, with CM=CN=4. Points J and K are on (AB)¯ so that (MJ)¯ and (NK)¯ are perpendicular to (AB)¯. What is the area of pentagon CMJKN? Express your answer as a common fraction.

[asy]
pair A,B,C,N,K,M,J;
A=(0,5);
B=(12,0);
C=(0,0);
N=(5,0);
M=(0,4);
J=(0.35,4.84);
K=(6,2.4);
draw(A--B--C--cycle,linewidth(0.7));
draw(N--K,linewidth(0.7));
draw(M--J,linewidth(0.7));
label("A",A,NW);
label("M",M,W);
label("C",C,SW);
label("N",N,S);
label("B",B,SE);
label("K",K,NE);
label("J",J,NE);
[/asy]
Original LaTeX notation
In $\triangle ABC$, $AB = 13$, $AC=5$ and $BC=12$.  Points $M$ and $N$ lie on $\overline{AC}$ and $\overline{BC}$, respectively, with $CM = CN = 4$. Points $J$ and $K$ are on $\overline{AB}$ so that $\overline{MJ}$ and $\overline{NK}$ are perpendicular to $\overline{AB}$. What is the area of pentagon $CMJKN$? Express your answer as a common fraction.

[asy]
pair A,B,C,N,K,M,J;
A=(0,5);
B=(12,0);
C=(0,0);
N=(5,0);
M=(0,4);
J=(0.35,4.84);
K=(6,2.4);
draw(A--B--C--cycle,linewidth(0.7));
draw(N--K,linewidth(0.7));
draw(M--J,linewidth(0.7));
label("$A$",A,NW);
label("$M$",M,W);
label("$C$",C,SW);
label("$N$",N,S);
label("$B$",B,SE);
label("$K$",K,NE);
label("$J$",J,NE);
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
a31fd718-8df1-581a-85c0-368d87dbb980
task_key
geometry--test--a31fd718-8df1-581a-85c0-368d87dbb980
task_revision_id
1
config / split
geometry / test

Official source

initial import