Week 14 Β· Regular polygons and symmetry; the coordinate plane, projections and the midpoint Β· 3 lessons
2
The coordinate plane: points, quadrants and projections on the axes
Mathematics Prep 1 Term 1, Unit 4, Lessons 5β6
GoalPlace points in the coordinate plane, name their quadrant and find their projections on the axes.
New words
quadrant Β· one of the four regions the axes cut the plane intoorigin Β· the point (0, 0) where the axes crossprojection Β· the point of an axis reached by going straight from the point to that axisordered pair Β· two numbers (x, y) that give the place of a point
Explanation
The x-axis is horizontal and the y-axis is vertical; they cross at the origin (0, 0). A point is written (x, y): first go along the x-axis, then up or down. The axes cut the plane into four quadrants: I has x > 0 and y > 0, II has x < 0 and y > 0, III has x < 0 and y < 0, and IV has x > 0 and y < 0. A point (x, 0) lies on the x-axis and a point (0, y) lies on the y-axis. Points on the axes belong to no quadrant. The projection of a point on the x-axis keeps x and sets y = 0. The projection on the y-axis keeps y and sets x = 0. The length of the projection of a segment on an axis is the positive difference of the coordinates.
Worked examples
Quadrants. A(3, 2): both positive, quadrant I. B(β4, 3): x negative, y positive, quadrant II. C(β2, β3): quadrant III. D(4, β2): quadrant IV. E(5, 0) is on the x-axis and F(0, β3) is on the y-axis, so they are in no quadrant.
Projection of a point. A is (2, 3). Its projection on the x-axis is (2, 0) and on the y-axis is (0, 3).
Projection of a segment. A(1, 4) and B(6, 2). On the x-axis the projection runs from (1, 0) to (6, 0), length |6 β 1| = 5. On the y-axis it runs from (0, 2) to (0, 4), length |4 β 2| = 2.
Area on the grid. A(1, 1), B(5, 1) and C(1, 4). AB lies on the line y = 1 and is 5 β 1 = 4 long. The height is the vertical side AC = 4 β 1 = 3. Area = Β½ Γ 4 Γ 3 = 6 square units.
An unknown. The point P(2k β 4, 3) is on the y-axis, so its x-coordinate is 0: 2k β 4 = 0, 2k = 4 and k = 2. So P is (0, 3).
Class activity
Treasure map. With a grown-up, draw axes on squared paper from β5 to 5. Hide a treasure at a point and give your partner clues: its quadrant, its projection on the x-axis and its projection on the y-axis.
Practice
1
Name the quadrant: (β5, 2), (β1, β6), (7, β3), (0, 4).
(β5, 2) is in II, (β1, β6) in III, (7, β3) in IV. (0, 4) is on the y-axis, in no quadrant.
2
Find the projections of B(β4, 3) on the x-axis and on the y-axis.
On the x-axis (β4, 0). On the y-axis (0, 3).
3
C(β3, 2) and D(4, β1). Find the lengths of the projections of CD on the two axes.
On the x-axis |4 β (β3)| = 7. On the y-axis |β1 β 2| = 3.
4
The point Q(k + 5, β2) is on the y-axis. Find k and Q.
k + 5 = 0, so k = β5 and Q is (0, β2).
5
Omar says the projection of (2, 3) on the x-axis is (0, 3). What is wrong?
On the x-axis the y-coordinate becomes 0 and x stays: (2, 0). (0, 3) is the projection on the y-axis.
