Extension & Challenge — Coordinates, Transformations & Angles — 2D Space & Structures: Transformations, Units, Area, Angles & the Cartesian Plane

Part A — The Cartesian Plane

The Cartesian plane has two number lines: the horizontal x-axis and the vertical y-axis. A point is written (x, y) — across first, then up or down. The axes divide the plane into four quadrants.

A four-quadrant coordinate grid with four points plotted: red, green, blue and purple. — Four points plotted on the Cartesian plane.

1 Write the coordinates of each plotted point.

Red: Green: Blue: Purple:

2 Which quadrant (1st, 2nd, 3rd or 4th) is each point in?

3 On the blank grid below, plot and label these points: P(−4, 3), Q(0, −2), R(5, 5), S(−1, −4).

A blank four-quadrant coordinate grid from minus 5 to 5 on both axes. — Plot your points here.

4 Three corners of a square are (1, 1), (1, 4) and (4, 4). What are the coordinates of the fourth corner?

Answer:

5 Reflect the point (3, 2) in the x-axis. What are the new coordinates?

Answer:

6 Reasoning. A point sits in the 1st quadrant. Its x and y values are equal and add to 8. What is the point?

Answer:

Part B — Transformations

There are three transformations that keep a shape the same size and shape (congruent): a translation (slide), a reflection (flip), and a rotation (turn).

A blue triangle P translated 4 units right to a green image triangle labelled P prime, with an arrow showing the slide. — Triangle P is translated (slid) 4 units right to P′.

1 Describe the transformation that maps triangle P onto triangle P′ above. Use the correct word and say how far and in which direction.

2 A vertex of triangle P is at (2, 3). After the slide of 4 units right, where is that vertex?

Answer:

3 Name the transformation in each case:

(a) a shape is flipped over a mirror line

(b) a shape is turned about a point

(c) a shape is slid without turning

4 After a transformation, a shape is the same size and the same shape but facing the opposite way. Which transformation was it most likely to be? Explain.

5 Reasoning. Which of translation, reflection and rotation keep a shape congruent (identical in size and shape)?

6 Open challenge. Write a set of instructions to move a shape from the bottom-left of a grid to the top-right using exactly two different transformations.

Part C — Angles

Three angle facts unlock most problems: angles on a straight line add to 180°, angles at a point add to 360°, and vertically opposite angles are equal.

Three angle diagrams: A shows 115 degrees and x on a straight line; B shows 47 degrees and y as vertically opposite angles; C shows 120, 130 and z meeting at a point. — Diagrams A, B and C for the angle challenges.

1 Find each missing angle in the diagram above. Give a reason for each.

(a) x = (angles on a line)

(b) y = (vertically opposite)

(c) z = (angles at a point)

2 Two angles sit on a straight line. One is 130°. What is the other?

Answer:

3 Four angles meet at a point. Three of them are 90°, 100° and 80°. Find the fourth.

Answer:

4 Classify each angle as acute, right, obtuse or reflex: 45°, 90°, 135°, 220°.

5 The three angles of a triangle are 60°, 70° and one unknown. Find the unknown angle.

Answer:

6 Prove it. Explain why vertically opposite angles are always equal. (Hint: both share an angle on a straight line.)