Question

1. pentagon mnpqr is shown on the coordinate grid. pentagon mnpqr is dilated with the origin as the center of dilation using the rule ((x,y)\to(kx,ky)) to create pentagon (mnpqr).

which statement is true?

• pentagon (mnpqr) is larger than pentagon (mnpqr), because the scale factor is greater than 1.
• pentagon (mnpqr) is smaller than pentagon (mnpqr), because the scale factor is less than 1.
• pentagon (mnpqr) is smaller than pentagon (mnpqr), because the scale factor is greater than 1.
• pentagon (mnpqr) is larger than pentagon (mnpqr), because the scale factor is less

Explanation:

Step1: Recall Dilation Rules

Dilation with scale factor \( k \): if \( k > 1 \), the image is larger than the original; if \( 0 < k < 1 \), the image is smaller. The rule here is \( (x,y) \to (kx,ky) \), and the selected option says the scale factor is less than 1, so the image (pentagon \( M'N'P'Q'R' \)) should be smaller.

Step2: Analyze Each Option

• First option: Claims scale factor > 1 (image larger), but the selected option's reasoning is scale factor < 1, so wrong.
• Second option: Matches dilation rule (scale factor < 1 → image smaller), correct.
• Third option: Contradicts dilation rule (scale factor > 1 would make image larger, not smaller), wrong.
• Fourth option: Contradicts dilation rule (scale factor < 1 makes image smaller, not larger), wrong.

Answer:

Pentagon \( M'N'P'Q'R' \) is smaller than pentagon \( MNPQR \), because the scale factor is less than 1.