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MATH / In rectangle ABCD, we have AB=8, BC=9, H is on (BC)¯ with BH=6, E is on AD with DE=4, line EC intersects line AH at G, and F is on line AD with (GF)¯⊥(AF)¯. Find the length (GF)¯. …

Problem

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

problem

In rectangle ABCDABCD, we have AB=8AB=8, BC=9BC=9, HH is on BC¯\overline{BC} with BH=6BH=6, EE is on ADAD with DE=4DE=4, line ECEC intersects line AHAH at GG, and FF is on line ADAD with GF¯⊥AF¯\overline{GF}\perp \overline{AF}. Find the length GF¯\overline{GF}. [asy] pair A,B,C,D,I,F,G,H; F=(0,0); D=(6,0); I=(10,0); A=(15,0); B=(15,8); H=(9,8); C=(6,8); G=(0,20); draw(A--B--C--D--cycle,linewidth(0.7)); draw(A--F--G--cycle,linewidth(0.7)); draw(G--I,linewidth(0.7)); label("4",(8,0),S); label("8",(15,4),E); label("6",(12,8),N); label("AA",A,SE); label("BB",B,NE); label("CC",C,W); label("DD",D,S); label("EE",I,S); label("FF",F,W); label("GG",G,W); label("HH",H,NE); [/asy]
Plain-text mathematical notation (without MathML)
In rectangle ABCD, we have AB=8, BC=9, H is on (BC)¯ with BH=6, E is on AD with DE=4, line EC intersects line AH at G, and F is on line AD with (GF)¯⊥(AF)¯. Find the length (GF)¯.

[asy]
pair A,B,C,D,I,F,G,H;
F=(0,0);
D=(6,0);
I=(10,0);
A=(15,0);
B=(15,8);
H=(9,8);
C=(6,8);
G=(0,20);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(A--F--G--cycle,linewidth(0.7));
draw(G--I,linewidth(0.7));
label("4",(8,0),S);
label("8",(15,4),E);
label("6",(12,8),N);
label("A",A,SE);
label("B",B,NE);
label("C",C,W);
label("D",D,S);
label("E",I,S);
label("F",F,W);
label("G",G,W);
label("H",H,NE);
[/asy]
Original LaTeX notation
In rectangle $ABCD$, we have $AB=8$, $BC=9$, $H$ is on $\overline{BC}$ with $BH=6$, $E$ is on $AD$ with $DE=4$, line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AD$ with $\overline{GF}\perp \overline{AF}$. Find the length $\overline{GF}$.

[asy]
pair A,B,C,D,I,F,G,H;
F=(0,0);
D=(6,0);
I=(10,0);
A=(15,0);
B=(15,8);
H=(9,8);
C=(6,8);
G=(0,20);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(A--F--G--cycle,linewidth(0.7));
draw(G--I,linewidth(0.7));
label("4",(8,0),S);
label("8",(15,4),E);
label("6",(12,8),N);
label("$A$",A,SE);
label("$B$",B,NE);
label("$C$",C,W);
label("$D$",D,S);
label("$E$",I,S);
label("$F$",F,W);
label("$G$",G,W);
label("$H$",H,NE);
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
10aa1b38-29c7-5139-9896-ec9a5fc08f40
task_key
geometry--test--10aa1b38-29c7-5139-9896-ec9a5fc08f40
task_revision_id
1
config / split
geometry / test

Official source

initial import