Question

ijklmnop is shown on the graph below. ijklmnop is dilated by a scale factor of 4 centered at the origin to create ijklmnop. what is the area of ijklmnop? write your answer as a whole number or as a decimal rounded to the nearest tenth.

Explanation:

Step1: Calculate the area of the original figure

The original figure \( IJKLMNOP \) can be divided into two rectangles.

• For the upper rectangle (with vertices \( I, J, K, L \)): The length \( IJ = 3 - 1=2 \) units (using \( x - \)coordinates: \( x_J - x_I=3 - 1 \)), and the width \( IK=9 - 7 = 2 \) units (using \( y - \)coordinates: \( y_K - y_I=9 - 7 \)). The area of this rectangle \( A_1=2\times2 = 4 \) square units.
• For the lower rectangle (with vertices \( P, O, N, M \)): The length \( PO=3 - 1 = 2 \) units (using \( x - \)coordinates: \( x_O - x_P=3 - 1 \)), and the width \( PN=3 - 2=1 \) unit (using \( y - \)coordinates: \( y_P - y_N=3 - 2 \)). The area of this rectangle \( A_2=2\times1=2 \) square units.

The total area of the original figure \( A_{original}=A_1 + A_2=4 + 2=6 \) square units.

Step2: Use the area - scale factor relationship

When a figure is dilated by a scale factor \( k \), the area of the dilated figure \( A_{dilated}\) is related to the area of the original figure \( A_{original}\) by the formula \( A_{dilated}=k^{2}\times A_{original}\).
Given \( k = 4 \), then \( A_{dilated}=4^{2}\times6\).

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Answer:

96