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MATH / Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the…

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problem

Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. The hypotenuse of the larger triangle is 16 centimeters. What is the number of centimeters in the length of the longer leg of the smaller triangle? [asy]size(150); pair O; for(int i = 2; i < 5; ++i){ draw(O--((2/sqrt(3))^i)*dir(30*i)); } for(int g = 2; g < 4; ++g){ draw( ((2/sqrt(3))^g)*dir(30*g)-- ((2/sqrt(3))^(g+1))*dir(30*g+30)); } label("16 cm", O--(16/9)*dir(120), W); //label("30∘30^{\circ}",.4*dir(0),dir(90)); //label("30∘30^{\circ}",.4*dir(25),dir(115)); label("30∘30^{\circ}",.4*dir(50),dir(140)); label("30∘30^{\circ}",.4*dir(85),dir(175)); real t = (2/(sqrt(3))); //draw(rightanglemark((1,.1),(1,0),(.9,0),s=3)); draw(rightanglemark(rotate(30)*(0,t**4),rotate(0)*(0,t**3),O,s=3)); draw(rightanglemark(rotate(0)*(0,t**3),rotate(-30)*(0,t**2),O,s=3)); //draw(rightanglemark(rotate(-30)*(0,t**2),rotate(-60)*(0,t**1),O,s=3)); [/asy]
Plain-text mathematical notation (without MathML)
Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. The hypotenuse of the larger triangle is 16 centimeters. What is the number of centimeters in the length of the longer leg of the smaller triangle?

[asy]size(150); pair O; for(int i = 2; i < 5; ++i){
draw(O--((2/sqrt(3))^i)*dir(30*i));
}
for(int g = 2; g < 4; ++g){
draw( ((2/sqrt(3))^g)*dir(30*g)-- ((2/sqrt(3))^(g+1))*dir(30*g+30));
}
label("16 cm", O--(16/9)*dir(120), W);
//label("30^(∘)",.4*dir(0),dir(90));
//label("30^(∘)",.4*dir(25),dir(115));
label("30^(∘)",.4*dir(50),dir(140));
label("30^(∘)",.4*dir(85),dir(175));
real t = (2/(sqrt(3)));
//draw(rightanglemark((1,.1),(1,0),(.9,0),s=3));
draw(rightanglemark(rotate(30)*(0,t**4),rotate(0)*(0,t**3),O,s=3));
draw(rightanglemark(rotate(0)*(0,t**3),rotate(-30)*(0,t**2),O,s=3));
//draw(rightanglemark(rotate(-30)*(0,t**2),rotate(-60)*(0,t**1),O,s=3));
[/asy]
Original LaTeX notation
Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. The hypotenuse of the larger triangle is 16 centimeters. What is the number of centimeters in the length of the longer leg of the smaller triangle?

[asy]size(150); pair O; for(int i = 2; i < 5; ++i){
draw(O--((2/sqrt(3))^i)*dir(30*i));
}
for(int g = 2; g < 4; ++g){
draw( ((2/sqrt(3))^g)*dir(30*g)-- ((2/sqrt(3))^(g+1))*dir(30*g+30));
}
label("16 cm", O--(16/9)*dir(120), W);
//label("$30^{\circ}$",.4*dir(0),dir(90));
//label("$30^{\circ}$",.4*dir(25),dir(115));
label("$30^{\circ}$",.4*dir(50),dir(140));
label("$30^{\circ}$",.4*dir(85),dir(175));
real t = (2/(sqrt(3)));
//draw(rightanglemark((1,.1),(1,0),(.9,0),s=3));
draw(rightanglemark(rotate(30)*(0,t**4),rotate(0)*(0,t**3),O,s=3));
draw(rightanglemark(rotate(0)*(0,t**3),rotate(-30)*(0,t**2),O,s=3));
//draw(rightanglemark(rotate(-30)*(0,t**2),rotate(-60)*(0,t**1),O,s=3));
[/asy]

level

Level 5

type

Prealgebra

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Source and history

task_id
1f7a00a0-aef5-56fc-a29c-10f6edbdc105
task_key
prealgebra--test--1f7a00a0-aef5-56fc-a29c-10f6edbdc105
task_revision_id
1
config / split
prealgebra / test

Official source

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