Distance between two points (Pythagoras)

Task goals

• Find the horizontal change, vertical change, and distance between two points plotted in the first quadrant, using a right-angled triangle diagram and Pythagoras' theory.
• Find the distance between two points where coordinates include negative values, recognising which Pythagorean triple applies.
• Find the distance between two points and show that it equals a given value, including cases with mixed positive and negative coordinates.
• Given one point, a known distance, and one missing coordinate of a second point, find a possible value of the missing coordinate.
• Identify and explain errors in an incorrect method for finding the distance between two points, such as adding the changes instead of combining them as a hypotenuse or forgetting to take a square root.
• Compare the lengths of two segments, or the distances of two points from a fixed point, to determine which is longer or closer.
• Solve harder distance problems involving shapes, such as the diagonal of a square or rectangle, constructing an integer-coordinate point a given distance from another point, or finding both possible values of a missing horizontal or vertical change when the total distance and one component are known.
• Reason about and verify distance results, including showing a distance is a whole number, testing a claimed distance, and giving two points that are an irrational distance apart.