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.

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

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

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

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

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

retangle with an inner circle tangent to three sides

answer: (A)
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If the length of the legs of a right triangle are $\,\sqrt{3}\;$ and $\;\sqrt{4}\,$, the length of the hypotenuse is:

$\,\sqrt{5}\,$

$\,\sqrt{7}\,$

$\,\sqrt{8}\,$

$\,\sqrt{9}\,$

$\,\sqrt{12}\,$

answer: (B)
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What is the vertical height (altitude) of the right cone whose radius of the circular base is equal to $\;\sqrt{3}\,$ centimeters and the slant height is 5 centimeters ?

answer:

Thinking:

The slant height of the cone is the distance from the apex to a point in the perimeter of the base. If the cone is a right cone, the length of all the slant heights are the same.

cone de geratriz 5cm e altura raiz de 3 cm

Solution:

$\,\left.\begin{array}{rcr} \mbox{slant height }\phantom{XXXX}\;\,\rightarrow\, & \;S_h\mbox{ = 5 cm }\; \\ \,\mbox{radius of the base}\phantom{XX} \rightarrow\, & R\,=\,\sqrt{3}\\ \mbox{Pythagorean theorem}\, \rightarrow\, & (S_h)^{\large 2}\,=\,H^{\large 2}\,+\,R^{\large 2}\; \\ \end{array} \right\}\;\Rightarrow\;$

$\;\Rightarrow\;5^{\large 2}\,=\,H^{\large 2}\,+\,(\sqrt{3})^{\large 2}\;\Leftrightarrow\;H\,=\,\sqrt{22} \mbox{ cm}$

the vertical height of the cone is $\,H\,=\,\sqrt{22}\,$ cm
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