Topic Review — Coordinate Geometry

Review Questions

1. Gradient and equations of lines. Fluency

   1. Find the gradient of the line passing through (2, 5) and (6, 13).
   2. Find the equation of the line with gradient −3 passing through (1, 4). Write in y = mx + b form.
   3. Find the equation of the line through (−1, 2) and (3, 10).
   4. State whether y = 5x − 2 and y = 5x + 7 are parallel, perpendicular, or neither.
   5. State the gradient of a line perpendicular to y = ¼x + 3.

2. x-intercepts, y-intercepts, and key features. Fluency

   1. Find the x-intercept and y-intercept of y = 3x − 9.
   2. A line passes through the origin with gradient 4. Write its equation.
   3. Find the equation of the horizontal line through (3, −7).
   4. Find the equation of the vertical line through (−5, 2).
   5. Find the equation of the line through (0, 6) parallel to y = −2x + 1.

3. Midpoint and distance. Fluency

   1. Find the midpoint of (4, −2) and (8, 6).
   2. Find the exact distance between (0, 0) and (8, 6).
   3. Find the exact distance between (−1, 3) and (5, −5). Leave as a surd.
   4. The midpoint of AB is (2, 5) and A is (0, 3). Find B.
   5. Find the midpoint of (−3, −4) and (7, 2).

4. Combining concepts. Understanding

   1. Show that the line joining A(1, 1) and B(5, 9) is parallel to the line joining C(0, −2) and D(3, 4).
   2. Find the equation of the perpendicular bisector of the segment joining P(2, 0) and Q(8, 4).
      1. First, find the midpoint M of PQ.
      2. Then find the gradient of PQ.
      3. Write the equation of the perpendicular bisector through M.
   3. Triangle ABC has vertices A(0, 0), B(6, 0), and C(3, 4). Find the length of the median from C to the midpoint of AB.
   4. Points P(1, 3), Q(5, 5), R(7, 1), S(3, −1) form a quadrilateral. Show it is a rhombus by showing all four sides are equal in length.

5. Mixed problem solving. Problem Solving

   1. A right-angled triangle has vertices at O(0, 0), A(4, 0), and B(0, 3).
      1. Find the length of the hypotenuse AB.
      2. Find the midpoint of the hypotenuse M.
      3. Show that M is equidistant from O, A, and B.
      4. Find the equation of the line through A and B.
   2. A map uses coordinates where 1 unit = 1 km. A hiker walks from campsite C(2, 3) to a lookout L(10, 9).
      1. What is the straight-line distance from C to L?
      2. A water source W is at the midpoint of CL. Find W’s coordinates.
      3. A ranger station at R(2, 9) is visible from the path CL. Find the equation of CL. Is R on the line?
   3. Two lines are defined: Line 1 through (0, 5) and (3, −1); Line 2 through (1, 4) perpendicular to Line 1.
      1. Find the equation of each line.
      2. Find their point of intersection by solving simultaneously.