Plane geometry

9000149406

Level: 
B
Given points \(A = [2;-5]\), \(B = [2;3]\) and \(C = [-4;-1]\), find the length of the altitude of the triangle \(ABC\) through the point \(C\). Hint: The altitude through the point \(C\) of a triangle \(ABC\) is the perpendicular line segment drawn from the vertex \(C\) to the line containing the side \(AB\).
\(6\)
\(\sqrt{2}\)
\(\frac{3} {2}\)
The points \(A\), \(B\), \(C\) do not define a triangle.

9000149409

Level: 
B
Find all lines which are parallel to \(p\colon x - 3y + 2 = 0\) and the distance from every of these lines to \(p\) is \(\sqrt{10}\).
\(p_{1}\colon x - 3y + 12 = 0\), \(p_{2}\colon x - 3y - 8 = 0\)
\(p\colon x - 3y = 0\)
\(p\colon x - 3y + \sqrt{10} = 0\)
\(p_{1}\colon x - 3y + \sqrt{10} = 0\), \(p_{2}\colon x - 3y -\sqrt{10} = 0\)

9000149410

Level: 
B
Find all lines passing through the point \(A = [-2;-6]\) such that the distance from the point \([0.0]\) to these lines is \(2\sqrt{2}\).
\(p_{1}\colon 7x + y + 20 = 0\), \(p_{2}\colon x - y - 4 = 0\)
\(p\colon 7x - y = 0\)
\(p\colon x + y + 2\sqrt{2} = 0\)
\(p_{1}\colon x - y + 2\sqrt{2} = 0\), \(p_{2}\colon x + y - 2\sqrt{2} = 0\)

9000151302

Level: 
B
Find the angle \(\varphi \) between the lines \(p\) and \(q\) given by their parametric equations. \[ p\colon \begin{aligned}[t] x& = 1 + 2t, & \\y& = 3 - 3t;\ t\in \mathbb{R}, \\ \end{aligned}\qquad q\colon \begin{aligned}[t] x& = 2 - k, & \\y& = 3 + k;\ k\in \mathbb{R} \\ \end{aligned} \]
\(11^{\circ }19'\)
\(88^{\circ }41'\)
\(45^{\circ }45'\)
\(54^{\circ }12'\)