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MATH / Circles with centers A and B have radii 3 and 8, respectively. A common internal tangent touches the circles at C and D, as shown. Lines AB and CD intersect at E, and AE=5. What is…

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

problem

Circles with centers AA and BB have radii 3 and 8, respectively. A common internal tangent touches the circles at CC and DD, as shown. Lines ABAB and CDCD intersect at EE, and AE=5AE = 5. What is CDCD? [asy] unitsize(0.2cm); pair A,B,C,D,E; A=(0,0); B=(18.3,0); E=(5,0); C=(1.8,-2.4); D=(13.5,6.4); draw(Circle(A,3),linewidth(0.7)); draw(Circle(B,8),linewidth(0.7)); draw(A--B,linewidth(0.7)); draw(C--D,linewidth(0.7)); label("AA",A,W); label("BB",B,S); label("CC",C,S); label("DD",D,NW); label("EE",E,S); [/asy]
Plain-text mathematical notation (without MathML)
Circles with centers A and B have radii  3 and 8, respectively. A common internal tangent touches the circles at C and D, as shown. Lines AB and CD intersect at E, and AE=5. What is CD?

[asy]
unitsize(0.2cm);
pair A,B,C,D,E;
A=(0,0);
B=(18.3,0);
E=(5,0);
C=(1.8,-2.4);
D=(13.5,6.4);
draw(Circle(A,3),linewidth(0.7));
draw(Circle(B,8),linewidth(0.7));
draw(A--B,linewidth(0.7));
draw(C--D,linewidth(0.7));
label("A",A,W);
label("B",B,S);
label("C",C,S);
label("D",D,NW);
label("E",E,S);
[/asy]
Original LaTeX notation
Circles with centers $A$ and $B$ have radii  3 and 8, respectively. A common internal tangent touches the circles at $C$ and $D$, as shown. Lines $AB$ and $CD$ intersect at $E$, and $AE = 5$. What is $CD$?

[asy]
unitsize(0.2cm);
pair A,B,C,D,E;
A=(0,0);
B=(18.3,0);
E=(5,0);
C=(1.8,-2.4);
D=(13.5,6.4);
draw(Circle(A,3),linewidth(0.7));
draw(Circle(B,8),linewidth(0.7));
draw(A--B,linewidth(0.7));
draw(C--D,linewidth(0.7));
label("$A$",A,W);
label("$B$",B,S);
label("$C$",C,S);
label("$D$",D,NW);
label("$E$",E,S);
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
2859d269-8c27-5d89-90d2-72b20c8ca5aa
task_key
geometry--test--2859d269-8c27-5d89-90d2-72b20c8ca5aa
task_revision_id
1
config / split
geometry / test

Official source

initial import