Question

graph the image of trapezoid stuv after a dilation with a scale factor of 3, centered at the origin.

Explanation:

Step1: Identify Coordinates

First, find the coordinates of each vertex of trapezoid \( STUV \):

• \( S(-3, -3) \) (assuming each grid square is 1 unit, looking at the graph: S is at x=-3, y=-3)
• \( T(3, -3) \) (T is at x=3, y=-3)
• \( U(3, 3) \) (U is at x=3, y=3)
• \( V(0, 3) \) (V is at x=0, y=3) Wait, correction: From the graph, V is at (0, 3)? Wait no, looking again: V is at (0, 3)? Wait the original graph: V is at (0, 3)? Wait the green dots: V is at (0, 3)? Wait no, the user's graph: V is at (0, 3)? Wait no, let's re-express:

Wait the original coordinates (from the graph):

• \( S \): Let's see, the x-coordinate: between -4 and -2, so -3; y-coordinate: -3 (since it's on y=-3). So \( S(-3, -3) \)
• \( T \): x=3, y=-3 (since it's on y=-3, x=3)
• \( U \): x=3, y=3 (since it's on y=3, x=3)
• \( V \): x=0, y=3 (since it's on y=3, x=0)

Wait, but in the graph, V is connected to U (x=3, y=3) with a horizontal line? Wait no, the original trapezoid: V is at (0, 3), U at (3, 3), T at (3, -3), S at (-3, -3)? Wait no, the graph shows S at (-3, -3)? Wait the user's graph: S is at (-3, -3)? Wait the grid: x from -10 to 10, y from -10 to 10. The green dots: S is at (-3, -3)? Wait no, looking at the graph: S is at (-3, -3)? Wait the coordinates:

Wait let's check again:

• \( V \): (0, 3) (since it's on the y-axis, x=0, y=3)
• \( U \): (3, 3) (x=3, y=3)
• \( T \): (3, -3) (x=3, y=-3)
• \( S \): (-3, -3) (x=-3, y=-3)

Yes, that makes a trapezoid with bases \( ST \) (length 6) and \( VU \) (length 3), and legs \( SV \) and \( TU \).

Step2: Apply Dilation (Scale Factor 3, Center at Origin)

Dilation centered at the origin with scale factor \( k \) transforms a point \( (x, y) \) to \( (k \cdot x, k \cdot y) \).

So for each vertex:

• \( S(-3, -3) \): Multiply by 3: \( S'(-3 \times 3, -3 \times 3) = (-9, -9) \)
• \( T(3, -3) \): Multiply by 3: \( T'(3 \times 3, -3 \times 3) = (9, -9) \)
• \( U(3, 3) \): Multiply by 3: \( U'(3 \times 3, 3 \times 3) = (9, 9) \)
• \( V(0, 3) \): Multiply by 3: \( V'(0 \times 3, 3 \times 3) = (0, 9) \)

Step3: Plot the New Coordinates

Now, plot the new points:

• \( S'(-9, -9) \)
• \( T'(9, -9) \)
• \( U'(9, 9) \)
• \( V'(0, 9) \)

Then connect the points \( S' \to T' \to U' \to V' \to S' \) to form the dilated trapezoid.

Answer:

To graph the dilated trapezoid \( S'T'U'V' \):

1. Find the original coordinates: \( S(-3, -3) \), \( T(3, -3) \), \( U(3, 3) \), \( V(0, 3) \).
2. Apply dilation (scale factor 3, center at origin):

• \( S' = 3 \times (-3, -3) = (-9, -9) \)
• \( T' = 3 \times (3, -3) = (9, -9) \)
• \( U' = 3 \times (3, 3) = (9, 9) \)
• \( V' = 3 \times (0, 3) = (0, 9) \)

1. Plot \( S'(-9, -9) \), \( T'(9, -9) \), \( U'(9, 9) \), \( V'(0, 9) \) and connect them to form the dilated trapezoid.