An Exploration in Rectangles

Transcript

An Exploration in RectanglesLet’s construct a rectangle ABCD with a length AB = 7 cm and a width BC = 4 cm. We imagine two points: Point X can move anywhere along the vertical side AD. Point Y can move anywhere along the vertical side BC. We try to find the relationship between the positions of X and Y and the length of the line segment XY that connects them. Let’s answer some questions At which positions will the points X and Y be at their closest? The points X and Y will be at their closest when the line segment XY is horizontal This is because Sides AD and BC are two parallel vertical lines. The shortest distance between any two parallel lines is a perpendicular line that connects them. In this case, any horizontal line (like AB) is perpendicular to both AD and BC Thus, we can many such cases When X is at point A, and Y is at point B When X is at point D, and Y is at point C When X is 1 cm down from A, and Y is 1 cm down from B How does the minimum distance between the points X and Y compare to the length of AB? The minimum distance of XY is exactly equal to the length of AB (which is 7 cm) When do you think they will be the farthest? They will be farthest when the line XY is a diagonal, stretching from a top corner on one side to a bottom corner on the other. Examples: When X is at point A, and Y is at point C (Diagonal AC) When X is at point D, and Y is at point B (Diagonal DB)