A

9000120302

Level: 
A
A cuboid has sides \(a = 5\, \mathrm{cm}\), \(b = 8\, \mathrm{cm}\), and \(c = \sqrt{111}\, \mathrm{cm}\). Find the length of the cuboid’s space diagonal \(u\) (see the picture).
\(10\sqrt{2}\, \mathrm{cm}\)
\(\sqrt{222}\, \mathrm{cm}\)
\(20\, \mathrm{cm}\)
\(2\sqrt{10}\, \mathrm{cm}\)
\(5\sqrt{7}\, \mathrm{cm}\)

9000120309

Level: 
A
The sides of a rectangular box shown in the picture are \(a = 3\, \mathrm{cm}\), \(b = 4\, \mathrm{cm}\), and \(c = 12\, \mathrm{cm}\). The space diagonal is \(u_{t}\) and the longest face diagonal is \(u_{s}\). Find the ratio \(u_{t} : u_{s}\).
\(13\sqrt{10} : 40\)
\(13 : \sqrt{153}\)
\(13 : 12\)
\(4\sqrt{10} : 5\)
\(4\sqrt{10} : 13\)

9000121705

Level: 
A
Consider an isosceles triangle \(ABC\) with sides \(AC\) and \(BC\) of equal length. The measure of the angle \( BAC\) is \(40^{\circ }\). \(X\) is the point of intersection between the line $AB$ and the line through the vertex \(C\) perpendicular to it. Find the measure of the angle \( BCX\).
\(50^{\circ }\)
\(80^{\circ }\)
\(100^{\circ }\)
\(40^{\circ }\)

9000120310

Level: 
A
The base of a rectangular box \(ABCDEFGH\) has sides \(|AB| = 6\, \mathrm{cm}\) and \(|BC| = 8\, \mathrm{cm}\). The angle between the solid diagonal \(AG\) and the base \(ABC\) is \(60^{\circ }\). Find the volume of the box.
\(480\sqrt{3}\, \mathrm{cm}^{3}\)
\(960\, \mathrm{cm}^{3}\)
\(288\sqrt{3}\, \mathrm{cm}^{3}\)
\(160\sqrt{3}\, \mathrm{cm}^{3}\)
\(240\, \mathrm{cm}^{3}\)

9000120307

Level: 
A
The lengths of a side, base diagonal and solid diagonal through the vertex \(A\) in a rectangular box \(ABCDEFGH\) are \(|AB| = 6\, \mathrm{cm}\), \(|AC| = 10\, \mathrm{cm}\), \(|AG| = 15\, \mathrm{cm}\). Find the volume of the box.
\(240\sqrt{5}\, \mathrm{cm}^{3}\)
\(900\, \mathrm{cm}^{3}\)
\(300\sqrt{5}\, \mathrm{cm}^{3}\)
\(600\sqrt{2}\, \mathrm{cm}^{3}\)
\(240\sqrt{2}\, \mathrm{cm}^{3}\)