Exercises

Precalculus for Everybody

Chapter 2. The Cartesian Plane and The Line

The Cartesian Plane

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

  1. \[ P = (0, 0)\text{, } \, Q = (1, 2) \]
  2. \[ P = (1, 3)\text{, } \, Q = (3, 5) \]
  3. \[ P = (-1, 1)\text{, } \, Q = (1, \sqrt{2}) \]
  4. Prove that the points \(A = (-2, 4)\), \(B = (-1, 3)\)   and   \(C = (2, -1)\) are collinear.
  5. If \(A = (-3, -5)\), \(M = (0, 2)\) and \(M\) is the midpoint of the segment \(\overline{AB}\), find \(B\).
  6. If \(B = (8, -12)\), \(M = (7/2, 3)\) and \(M\) is the midpoint of the segment \(\overline{AB}\), find \(A\).
  7. Prove that \(A = (2, -3)\), \(B = (4, 2)\) and \(C = (-1, 4)\) are the vertices of an isosceles triangle.
  8. Prove that \(A = (4, 1)\), \(B = (2, 2)\) and \(C = (-1, -4)\) are the vertices of a right triangle.
  9. Prove that \(A = (1, 2)\), \(B = (4, 8)\), \(C = (5, 5)\) and \(D = (2, -1)\) are the vertices of a parallelogram.
  10. Prove that \(A = (0, 2)\), \(B = (1, 1)\), \(C = (2, 3)\) and \(D = (-1, 0)\) are the vertices of a rhombus.
  11. Prove that \(A = (1, 1)\), \(B = (11, 3)\), \(C = (10, 8)\) and \(D = (0, 6)\) are the vertices of a rectangle.
  12. Prove that \(A = (-4, 1)\), \(B = (1, 3)\), \(C = (3, -2)\) and \(D = (-2, -4)\) are the vertices of a square.
  13. Find the points \(P = (x, 2)\) whose distance to the point \((-1, -2)\) is 5 units.
  14. Find the points \(P = (1, y)\) whose distance to the point \((-4, 1)\) is 13 units.
  15. 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 \(B = (-4, -3)\).
  16. 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.
  17. The midpoints of the sides of a triangle are \(M = (2, -1)\), \(N = (-1, 4)\) and \(Q = (-2, 2)\). Find the vertices.
  18. 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.
  19. 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.
    1. \[ \sqrt{5}, \; \left( \frac{1}{2}, \, 1 \right) \]
       
    2. \[ 2 \sqrt{2}, \; (2, \, 4) \]
       
    3. \[ \sqrt{ 7 – 2 \sqrt{2}}, \; \left( 0, \, \frac{ 1 + \sqrt{2} }{2}\right) \]
       
    1. \[ B = (3, \, 9) \]
       
    2. \[ A = (-1, \, 18) \]
       
    1. \[ (2, \, 2) \quad \text{ and } \quad (-4, \, 2) \]
       
    2. \[ (1, \, 13) \quad \text{ and } \quad (1, \, -11) \]
       
    3. \[ 5x + 2y – 3 = 0 \]
       
    4. \[ x^2 + y^2 = 9 \]
       
    5. \[ (1, \, -3), \, (3, \, 1), \, (-5, \, 7) \]
       
    6. \[ (-2, \, -5), \, (0, \, -9) \]
    7. \[ \left( \frac{9}{2}, \, 1 \right) \]