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MATH / Two parallel chords in a circle have lengths 10 and 14, and the distance between…

Problem

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

problem

Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length a\sqrt{a}. Find the value of aa. [asy] import olympiad; import geometry; size(100); defaultpen(linewidth(0.8)); draw(unitcircle); draw(Label("14",align=N),dir(30)--dir(150)); draw(Label("10",align=N),dir(-40)--dir(-140)); draw(Label("a\sqrt{a}",align=N),dir(-5)--dir(-175)); distance(rotate(90)*"6",(1,Sin(30)),(1,Sin(-40)),-9,Arrows(size=1)); [/asy]
Plain-text mathematical notation (without MathML)
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length √(a). Find the value of a. [asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
draw(unitcircle);
draw(Label("14",align=N),dir(30)--dir(150));
draw(Label("10",align=N),dir(-40)--dir(-140));
draw(Label("√(a)",align=N),dir(-5)--dir(-175));
distance(rotate(90)*"6",(1,Sin(30)),(1,Sin(-40)),-9,Arrows(size=1));

[/asy]
Original LaTeX notation
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length $\sqrt{a}$. Find the value of $a$. [asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
draw(unitcircle);
draw(Label("14",align=N),dir(30)--dir(150));
draw(Label("10",align=N),dir(-40)--dir(-140));
draw(Label("$\sqrt{a}$",align=N),dir(-5)--dir(-175));
distance(rotate(90)*"6",(1,Sin(30)),(1,Sin(-40)),-9,Arrows(size=1));

[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
8179e6a1-1a6d-59ee-825a-64a93d613e0b
task_key
geometry--test--8179e6a1-1a6d-59ee-825a-64a93d613e0b
task_revision_id
1
config / split
geometry / test

Official source

initial import