Lines and planes: distances and angles

9000128802

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). The point \(M\) is the middle of the side \(CV \). Find the distance between the point \(M\) and the line \(BC\).
\(\frac{5} {2}\, \mathrm{cm}\)
\(\frac{\sqrt{34}} {2} \, \mathrm{cm}\)
\(\frac{\sqrt{7}} {2} \, \mathrm{cm}\)

9000128803

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). The point \(M\) is the middle of the side \(CV \). Find the distance between the point \(M\) and the line \(AD\).
\(\frac{\sqrt{97}} {2} \, \mathrm{cm}\)
\(\frac{\sqrt{106}} {2} \, \mathrm{cm}\)
\(\frac{\sqrt{65}} {2} \, \mathrm{cm}\)

9000128804

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). Find the distance between the line \(AD\) and the plane \(BCV \).
\(\frac{24} {5} \, \mathrm{cm}\)
\(\frac{15\sqrt{34}} {5} \, \mathrm{cm}\)
\(6\, \mathrm{cm}\)

9000128805

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). Find the angle between the line \(BV \) and the plane \(ABC\). Round to two decimal places.
\(43.31^{\circ }\)
\(59.04^{\circ }\)
\(45^{\circ }\)

9000128806

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). The point \(M\) is the middle of the side \(CV \). Find the angle between the line \(AM\) and the plane \(ABC\). Round to two decimal places.
\(17.45^{\circ }\)
\(34.50^{\circ }\)
\(18.32^{\circ }\)

9000128801

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). The point \(M\) is the middle of the side \(CV \). Find the distance between the point \(M\) and the plane \(ABC\).
\(2\, \mathrm{cm}\)
\(\frac{\sqrt{34}} {2} \, \mathrm{cm}\)
\(\frac{5} {2}\, \mathrm{cm}\)

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 \(u_{t} : u_{s}\).
\(13\sqrt{10} : 40\)
\(13 : \sqrt{153}\)
\(13 : 12\)
\(4\sqrt{10} : 5\)
\(4\sqrt{10} : 13\)