Lista de exercícios do ensino médio para impressão

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Find the lateral area, the total area and the volume of a circular cone that is circumscribed to a sphere of radius $\,r\,$ and whose axial cut is an equilateral triangle.

answer: $\,S_{\text lat}\,=\,6\,\pi\,r^2\,$; $\,S_{\text total}\,=\,9\,\pi\,r^2\,$; $\,V_{\text olume}\,=\,3\,\pi\,r^3\,$

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In the following picture:

$\,\overline{PP'}\,$ is the diameter of the sphere whose center is $\,O\,$, $\;M\,$ é is the center of a intersection with a plane perpendicular to $\,\overline{PP'}\,$. Also the measures are $\,\overline{AP}\,=\,6\,cm\;$ and $\,\overline{AP'}\,=\,8\,cm\;$. Calculate the area of the cicle whose center is $\,M\,$.

$\,\overline{PP'}\,$ is the diameter of the sphere whose center is $\,O\,$, $\;M\,$ é is the center of a intersection with a plane perpendicular to $\,\overline{PP'}\,$. Also the measures are $\,\overline{AP}\,=\,6\,cm\;$ and $\,\overline{AP'}\,=\,8\,cm\;$. Calculate the area of the cicle whose center is $\,M\,$.

In the picture the line r is parallel to the line s . Find the value of $\,\hat{\,\alpha\,}\,$.

Knowing that in the picture r // s, the value of $\;\hat{\;x\;}\;$ is:

a)

90°

b)

100°

c)

110°

d)

120°

e)

None of the above.

Given the following picture, find the value of $\;\hat{\;x\;}\;$ :

In the given picture, if r // s , we can say that the value of $\,\hat{\,\alpha\,}\,$ is:

a)

90°

b)

100°

c)

110°

d)

120°

e)

22°40'

Plot in the following xOy plane the points A(4; 3) , B(-2; 5) , C(-4; -2) , D(3; -4) , E(2; 0) , F(0; -3) , G$(\frac{3}{2};\;\frac{5}{2})\;$ e H($-\frac{1}{2}$; -4).

The lantern shown in the picture is suspended by two strings perpendicular to each other. Knowing the length of each string is 1/2 and 6/5 , respectively, what is the distance between the lantern and the ceiling?

a)

1,69

b)

1,3

c)

0,6

d)

1/2

e)

6/13

In the picture below, ABCD is a rectangle, $\,\overline{AB}\,=\,4\,$, $\,\overline{BC}\,=\,1\;$ and $\,\overline{DE}\,=\,\overline{EF}\,=\,\overline{FC}\;$. The length of $\,\overline{BG}\,$ is:

a)

$\,\dfrac{\sqrt{5}}{4}\,$

b)

$\,\dfrac{5}{2}\,$

c)

$\,\dfrac{9}{4}\,$

d)

$\,\dfrac{11}{4}\,$

e)

$\,\dfrac{5}{\sqrt{2}}\,$

In the picture below, ABFG and BCDE are squares and their each one sides measures, respectively, a and b. Knowing that $\;\overline{AG}\,=\,\overline{CD}\,+\,2\;\,$ and the perimeter of the triangle ACG is 12, we can assure that a e b are both inside the interval:

a)

]1; 5[

b)

]0; 4[

c)

]2; 6[

d)

]3; 7[

e)

]4; 8[

Knowing that in the right triangle ABC shown in the picture b = 1 and c = 2, find x.

a)

$\,\sqrt{2}\,$

b)

$\,\dfrac{3}{2}\,$

c)

$\,\dfrac{3\sqrt{2}}{2}\,$

d)

$\,\dfrac{2}{3}\,$

e)

$\,\dfrac{2\sqrt{2}}{3}\,$

(FUVEST - 1977)

Given that:

$\,\overline{MP}\;\bot\;s\,$;$\;\overline{MQ}\;\bot\;t\,$;$\;\overline{MQ}\;\bot\;\overline{PQ}\,$;$\;\overline{MP}\,=\,6$

The length of $\,\overline{PQ}\,$ é is:

$\,\overline{MP}\;\bot\;s\,$;$\;\overline{MQ}\;\bot\;t\,$;$\;\overline{MQ}\;\bot\;\overline{PQ}\,$;$\;\overline{MP}\,=\,6$

The length of $\,\overline{PQ}\,$ é is:

a)

$\,3\sqrt{3}\,$

b)

$\,3\,$

c)

$\,6\sqrt{3}\,$

d)

$\,4\sqrt{3}\,$

e)

$\,2\sqrt{3}\,$

The picture shows the rhombus ABCD and the point A is the center of the circle that has a 4 cm length radius. Find the area of the rhombus in square centimeters.

a)

$\,4\sqrt{3}\,$

b)

$\,8\,$

c)

$\,12\,$

d)

$\,8\sqrt{3}\,$

e)

$\,12\sqrt{3}\,$

(FESP - 1991) An equilateral triangle ABC is inscriibed in a circle whose radius is 6 cm length. The triangle is intercepted by a diameter MN of the circle, forming a trapezoid, as shown in the picture below. We can say that the quotient of the triangle ABC area divided by the trapezoid APQC area is:

a)

$\,\dfrac{5}{4}\,$

b)

$\,\dfrac{9}{5}\,$

c)

$\,\dfrac{9}{8}\,$

d)

$\,\dfrac{9}{4}\,$

e)

$\,\dfrac{8}{5}\,$

AB is the diameter of the circle whose center is O and the triangle ABC is inscribed in. The quotient $\,\dfrac{s}{S}\,$ where the area $\,s\,$ of the triangle ACO is divided by the area $\,S\,$ of the triangle COB is:

a)

$\,\dfrac{5}{4}\,$

b)

$\,\dfrac{4}{3}\,$

c)

$\,\dfrac{3}{4}\,$

d)

$\,1\,$

e)

$\,\dfrac{\sqrt{3}}{2}\,$

The circle as shown in the picture below, with center P and radius 2, is tangent to three sides of the rectangle ABCD. Given that the total area of the rectangle is 32, find the distance between the point P and the diagonal AC.

a)

$\,2\dfrac{\sqrt{5}}{5}\,$

b)

$\,\dfrac{\sqrt{5}}{2}\,$

c)

$\,\dfrac{\sqrt{5}}{5}\,$

d)

$\,2\sqrt{5}\,$

e)

$\,3\dfrac{\sqrt{5}}{5}\,$

Knowing that O is the center of the circle, find x in the following cases:

a)

b)

The circles in the picture are externally tangent. The distance between the centers $\,\overline{OC}\,$ is 28 cm and the difference between their radii is 8 cm. Find the length of each radius.

Find x in the following cases:

a) s is perpendicular to $\;\overline{AB}\,$

b) $\,\overline{PA}\,$ and $\,\overline{PB}\,$ are tangent segments to the circle

The radius of the circle C in the picture is 16 cm and the point P is located 7 cm far from the center O.

What is the distance between P and the circunference of the circle C?

What is the radius of the following circle which center is O,

given that:

AB = 3x - 3 and

OA = x + 3.

AB = 3x - 3 and

OA = x + 3.

The length of the arc $\,\stackrel \frown{AB}\,$, as in the picture, is 22 cm and O is the center of the circle. The perimeter of the circle (circumference) is:

a)

990 cm

b)

67 cm

c)

176 cm

d)

88 cm

e)

none of these answers

M is the midpoint of AB. Find $\,\overline{AB}\,$ length.

a)

b)

The length of AB is 17 cm. Find x in the following cases:

a)

b)

c)

d)

With M being the midpoint of $\,\overline{AB}\,$, find x:

a)

b)

In the cases a) and b), knowing that AB = 31, what is the length of PQ?

a)

b)