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# equilateral triangle

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An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction.

Oct 16, 2019

#1
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An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction.

Ein gleichseitiges Dreieck hat Seiten mit einer Länge von 8 Einheiten. Ein gleichseitiges Dreieck mit 4 Einheiten langen Seiten wird oben abgeschnitten, wobei ein gleichschenkliges Trapez verbleibt. Wie ist das Verhältnis der Fläche des kleineren Dreiecks zur Fläche des Trapezes?

Hello Guest!

$$x=\frac{A_4}{A_8-A_4}=\frac{\frac{4^2\sqrt{3}}{4}}{\frac{8^2\sqrt{3}}{4}-\frac{4^2\sqrt{3}}{4}}\\ x=\frac{4\sqrt{3}}{16\sqrt{3}-4\sqrt{3}}=\frac{4\sqrt{3}}{12\sqrt{3}}\\$$

$$x=\frac{1}{3}$$

The ratio of the area of the smaller triangle to the area of the trapezoid is $$\frac{1}{3}$$.

!

Oct 16, 2019
edited by asinus  Oct 16, 2019
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Area of   large triangle   =   (1/2)*8^2 * sin (60)  =   32 sin (60)  (1)

Area of small  triangle  =  (1/2) 4^2  * sin (60)   =   8 sin (60)    (2)

Area of trapezoid  =  (1) - (2)  =     24 sin (60)     (3)

Areae of small triangle to trapezoid  =   (2)  / (3)    =    8 / 24  =   1 / 3

Oct 17, 2019