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MATH / In acute triangle ABC, altitudes AD, BE, and CF intersect at the orthocenter H. If BD=5, CD=9, and CE=42/5, then find the length of HE. [asy] unitsize(0.3 cm); pair A, B, C, D, E, …

Problem

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

problem

In acute triangle ABCABC, altitudes ADAD, BEBE, and CFCF intersect at the orthocenter HH. If BD=5BD = 5, CD=9CD = 9, and CE=42/5CE = 42/5, then find the length of HEHE. [asy] unitsize(0.3 cm); pair A, B, C, D, E, F, H; A = (5,12); B = (0,0); C = (14,0); D = (A + reflect(B,C)*(A))/2; E = (B + reflect(C,A)*(B))/2; F = (C + reflect(A,B)*(C))/2; H = extension(B,E,C,F); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); label("AA", A, N); label("BB", B, SW); label("CC", C, SE); label("DD", D, S); label("EE", E, NE); label("FF", F, NW); label("HH", H, SE); [/asy]
Plain-text mathematical notation (without MathML)
In acute triangle ABC, altitudes AD, BE, and CF intersect at the orthocenter H.  If BD=5, CD=9, and CE=42/5, then find the length of HE.

[asy]
unitsize(0.3 cm);

pair A, B, C, D, E, F, H;

A = (5,12);
B = (0,0);
C = (14,0);
D = (A + reflect(B,C)*(A))/2;
E = (B + reflect(C,A)*(B))/2;
F = (C + reflect(A,B)*(C))/2;
H = extension(B,E,C,F);

draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
draw(C--F);

label("A", A, N);
label("B", B, SW);
label("C", C, SE);
label("D", D, S);
label("E", E, NE);
label("F", F, NW);
label("H", H, SE);
[/asy]
Original LaTeX notation
In acute triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$.  If $BD = 5$, $CD = 9$, and $CE = 42/5$, then find the length of $HE$.

[asy]
unitsize(0.3 cm);

pair A, B, C, D, E, F, H;

A = (5,12);
B = (0,0);
C = (14,0);
D = (A + reflect(B,C)*(A))/2;
E = (B + reflect(C,A)*(B))/2;
F = (C + reflect(A,B)*(C))/2;
H = extension(B,E,C,F);

draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
draw(C--F);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, NW);
label("$H$", H, SE);
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
131dba59-f88c-5736-bbbe-550bd1e3a76a
task_key
geometry--test--131dba59-f88c-5736-bbbe-550bd1e3a76a
task_revision_id
1
config / split
geometry / test

Official source

initial import