A rectangle has length 40cm and width 16cm. Point M is the midpoint of one side. The paper is folded along a line through M. If a vertex of the side containing M does not land on the opposite side after folding, find the length of the crease (in cm). Fill in the blank: .
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Step-by-Step Solution
Step 1: Define Coordinates and Points
Let's set up a coordinate system for the rectangle. We place vertex D at the origin (0,0). The length of the rectangle is 40cm and the width is 16cm. So the vertices are A(0,16), B(40,16), C(40,0), and D(0,0). Point M is the midpoint of side CD, so its coordinates are (20,0).
Step 2: Determine Folded Vertex Position
When the paper is folded along a line through M, vertex D moves to a new position, let's call it D′. The problem states that D′ does not land on the opposite side AB. This means D′ must lie on the side BC or AD, or outside the rectangle. Since M is on CD, and we are folding a vertex from CD, D′ will be on BC or AD. Given the setup, D′ will land on side BC. Let D′=(x,y).
Step 3: Apply Properties of Folding
Folding implies that the distance from M to D is equal to the distance from M to D′. Since M=(20,0) and D=(0,0), MD=20cm. Therefore, MD′=20cm. Since D′ lies on side BC, its x -coordinate must be 40. So D′=(40,y).
Step 4: Calculate y -coordinate of D′
Using the distance formula for MD′, we have (40−20)2+(y−0)2=202. This simplifies to 202+y2=202, which means y2=0, so y=0. This implies D′ is at (40,0), which is point C.
Step 5: Determine the Crease Line
The crease line is the perpendicular bisector of the segment connecting the original point D and its folded position D′. Since D=(0,0) and D′=(40,0), the segment DD′ lies on the x -axis. The midpoint of DD′ is (20,0), which is M. The perpendicular bisector of DD′ is a vertical line x=20. However, the crease must pass through M. This means the crease is the line segment from M to a point on the top side AB. Let the crease be MN, where N is on AB.
Step 6: Find the Length of the Crease
Since the crease is the line segment MN where M=(20,0) and N is on AB with x -coordinate 20, N=(20,16). The length of the crease MN is the distance between (20,0) and (20,16), which is 16−0=16cm.