Question

dilate rectangle qrst by a scale factor of 3 with the center of dilation at point q.

Explanation:

Step1: Determine the distance from Q to other points

• Let's assume the grid has unit length.
• The horizontal distance from \(Q\) to \(T\) is \(2\) units, and the vertical distance from \(Q\) to \(R\) is \(1\) unit.
• The horizontal distance from \(Q\) to \(S\) (in the x - direction) is \(2\) units, and the vertical distance from \(Q\) to \(S\) (in the y - direction) is \(1\) unit.

Step2: Apply the scale factor

• For a dilation with scale factor \(k = 3\) and center \(Q\):
• If a point \(P(x,y)\) (relative to \(Q\)) has coordinates \((x,y)\), the dilated point \(P^{\prime}(x^{\prime},y^{\prime})\) has coordinates \((kx,ky)\) (relative to \(Q\)).
• For point \(T\) (relative to \(Q\), \(x = 2,y = 0\)), after dilation, the new \(x\) - coordinate (relative to \(Q\)) is \(x^{\prime}=3\times2 = 6\), and \(y^{\prime}=3\times0=0\).
• For point \(R\) (relative to \(Q\), \(x = 0,y = 1\)), after dilation, the new \(y\) - coordinate (relative to \(Q\)) is \(y^{\prime}=3\times1 = 3\), and \(x^{\prime}=3\times0 = 0\).
• For point \(S\) (relative to \(Q\), \(x = 2,y = 1\)), after dilation, \(x^{\prime}=3\times2=6\) and \(y^{\prime}=3\times1 = 3\).

Step3: Plot the new points

• Since \(Q\) is the center of dilation, \(Q\) remains in the same position.
• Plot the new points:
• Start at \(Q\). Move \(6\) units to the right (from \(Q\)) to get the new \(T^{\prime}\).
• Move \(3\) units up (from \(Q\)) to get the new \(R^{\prime}\).
• Move \(6\) units to the right and \(3\) units up (from \(Q\)) to get the new \(S^{\prime}\).
• Connect the points \(Q\), \(R^{\prime}\), \(S^{\prime}\), \(T^{\prime}\) to form the dilated rectangle.

Answer:

Follow the above steps to draw the dilated rectangle with \(Q\) as the center, scale factor \(3\). The new rectangle will have sides that are \(3\) times the length of the original rectangle \(QRST\) (relative to the center \(Q\)).