Question

dilate the line segment by a scale factor of 2 with a center of dilation at the origin. draw the dilated segment on the provided graph.
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Explanation:

Step1: Identify Original Points

The original line segment has endpoints. Let's find their coordinates. From the graph, one endpoint is at \((0, 0)\) (origin) and the other is at \((4, 2)\) (wait, no, let's check axes: x - axis is vertical? Wait, maybe I mixed axes. Wait, the x - axis is going down, y - axis to the right? Wait, no, standard graph: x - horizontal, y - vertical. Wait, in the given graph, the x - axis is vertical (going down) and y - axis is horizontal (going right). So the first point (origin - like) is at \((x = 0, y = 0)\) (since x - axis starts at top with 0, goes down to 16; y - axis starts at left with 0, goes right to 16). The other endpoint: let's count the grid. From the origin (0,0), moving down (x - direction) 4 units and right (y - direction) 2 units? Wait, no, maybe the coordinates are (x, y) where x is vertical (down) and y is horizontal (right). So first point: \((0, 0)\), second point: \((4, 2)\)? Wait, no, looking at the segment: from (0,0) (x=0, y=0) to (4, 2) in x - y (x down, y right). Wait, dilation with scale factor 2 and center at origin.

Step2: Apply Dilation Formula

The dilation of a point \((x,y)\) with center at origin \((0,0)\) and scale factor \(k\) is \((kx, ky)\). So for the first point \((0,0)\), after dilation, it remains \((0,0)\) (since \(0\times2 = 0\), \(0\times2 = 0\)). For the second point, let's find its original coordinates. Wait, maybe I got the axes wrong. Let's re - examine: the x - axis is labeled with 0 at the top, then 2,4,...16 going down. The y - axis is labeled with 0 at the left, then 2,4,...16 going right. So the first endpoint is at (x = 0, y = 0) (top - left corner of the segment's start). The second endpoint: moving down 4 units (x = 4) and right 2 units (y = 2)? Wait, no, the segment goes from (0,0) (x=0, y=0) to (4, 2) in x - y (x down, y right). Wait, no, the length: let's count the grid squares. From (0,0) to (4, 2) (x - 4, y - 2). Then dilation with scale factor 2: new coordinates are (0×2, 0×2)=(0,0) and (4×2, 2×2)=(8,4). Wait, no, maybe the original second point is (2,4)? Wait, I think I mixed x and y. Let's look at the segment: the horizontal (y) distance from origin is 2 units? No, the segment is from (0,0) (x=0, y=0) to (2,4) (x=4? No, the grid lines: each grid square is 1 unit. Let's count the number of grid squares. The segment goes from (0,0) (x=0, y=0) to (4, 2) in x - y (x down, y right). Wait, maybe the correct original coordinates are (0,0) and (2,4) (x=4? No, I'm confused. Wait, dilation: center at origin, scale factor 2. So each coordinate (x,y) becomes (2x, 2y). Let's assume the original endpoints are \((0,0)\) and \((2,4)\) (maybe I had x and y reversed). If x is horizontal (right) and y is vertical (down), then the first point is (0,0), the second is (2,4). Then dilation: (0×2, 0×2)=(0,0) and (2×2, 4×2)=(4,8). Wait, maybe the axes are standard: x - horizontal (right), y - vertical (up). But in the graph, x - axis is going down (so y - axis is up? No, the labels: x - axis has 0 at the top, then 2,4,...16 going down (so x increases downward), y - axis has 0 at the left, 2,4,...16 going right (y increases rightward). So a point's coordinates are (x, y) where x is downward distance, y is rightward distance. So original segment: one endpoint at (0,0) (x=0, y=0), the other at (4, 2) (x=4, y=2). Dilation with scale factor 2: new endpoints are (0×2, 0×2)=(0,0) and (4×2, 2×2)=(8,4). So we need to draw a segment from (0,0) to (8,4) on the graph.

Answer:

To dilate the line segment with scale factor 2 and center at the origin:

1. Identify the endpoints of the original segment. Let the original endpoints be \(A(0,0)\) and \(B(4,2)\) (assuming x - downward, y - rightward).
2. Apply the dilation formula \((x,y)\to(2x,2y)\). For \(A(0,0)\), the image \(A'\) is \((0,0)\). For \(B(4,2)\), the image \(B'\) is \((8,4)\).
3. Draw the line segment connecting \(A'(0,0)\) and \(B'(8,4)\) on the graph.

(Note: The actual drawing would involve plotting the points \((0,0)\) and \((8,4)\) (with x - downward and y - rightward axes) and drawing the segment between them.)