In exercises, from 1 to 3, find the distance between the points \(\boldsymbol{P}\) and \(\boldsymbol{Q}\), and find the midpoint of the segment \(\boldsymbol{\overline{PQ}}\).

\[ P = (0, 0)\text{, } \, Q = (1, 2) \]

\[ P = (1, 3)\text{, } \, Q = (3, 5) \]

\[ P = (1, 1)\text{, } \, Q = (1, \sqrt{2}) \]
 Prove that the points \(A = (2, 4)\), \(B = (1, 3)\) and \(C = (2, 1)\) are collinear.
 If \(A = (3, 5)\), \(M = (0, 2)\) and \(M\) is the midpoint of the segment \(\overline{AB}\), find \(B\).
 If \(B = (8, 12)\), \(M = (7/2, 3)\) and \(M\) is the midpoint of the segment \(\overline{AB}\), find \(A\).
 Prove that \(A = (2, 3)\), \(B = (4, 2)\) and \(C = (1, 4)\) are the vertices of an isosceles triangle.
 Prove that \(A = (4, 1)\), \(B = (2, 2)\) and \(C = (1, 4)\) are the vertices of a right triangle.
 Prove that \(A = (1, 2)\), \(B = (4, 8)\), \(C = (5, 5)\) and \(D = (2, 1)\) are the vertices of a parallelogram.
 Prove that \(A = (0, 2)\), \(B = (1, 1)\), \(C = (2, 3)\) and \(D = (1, 0)\) are the vertices of a rhombus.
 Prove that \(A = (1, 1)\), \(B = (11, 3)\), \(C = (10, 8)\) and \(D = (0, 6)\) are the vertices of a rectangle.
 Prove that \(A = (4, 1)\), \(B = (1, 3)\), \(C = (3, 2)\) and \(D = (2, 4)\) are the vertices of a square.
 Find the points \(P = (x, 2)\) whose distance to the point \((1, 2)\) is 5 units.
 Find the points \(P = (1, y)\) whose distance to the point \((4, 1)\) is 13 units.
 Find an equation of two variables, \(x\) and \(y\), which is satisfied by the coordinates of all the points \(P = (x, y)\) equidistant from the points \(A = (6, 1)\) and \(B = (4, 3)\).
 Find an equation of two variables \(x\) and \(y\), which is satisfied by the coordinates of all the points \(P = (x, y)\) whose distance to the origin is 3 units.
 The midpoints of the sides of a triangle are \(M = (2, 1)\), \(N = (1, 4)\) and \(Q = (2, 2)\). Find the vertices.
 Two adjacent vertices of a parallelogram are \(A = (2, 3)\) and \(B = (4, 1)\). If the the point \(M = (1, 3)\) bisects the diagonals, find the other two vertices.
 The vertices of a quadrilateral are \(A = (2, 14)\), \(B = (3, 4)\), \(C = (6, 2)\) and \(D = (6, 6)\). Find the intersection point of the diagonals.
Answers


\[ \sqrt{5}, \; \left( \frac{1}{2}, \, 1 \right) \]

\[ 2 \sqrt{2}, \; (2, \, 4) \]

\[ \sqrt{ 7 – 2 \sqrt{2}}, \; \left( 0, \, \frac{ 1 + \sqrt{2} }{2}\right) \]



\[ B = (3, \, 9) \]

\[ A = (1, \, 18) \]



\[ (2, \, 2) \quad \text{ and } \quad (4, \, 2) \]

\[ (1, \, 13) \quad \text{ and } \quad (1, \, 11) \]

\[ 5x + 2y – 3 = 0 \]

\[ x^2 + y^2 = 9 \]

\[ (1, \, 3), \, (3, \, 1), \, (5, \, 7) \]

\[ (2, \, 5), \, (0, \, 9) \]

\[ \left( \frac{9}{2}, \, 1 \right) \]
