Question

Triangle MNO and triangle PQR are shown on the coordinate plane.

A triangle on the coordinate plane with vertices M at 1 comma 1, N at 1 comma 3, and O at 4 comma 3. A second triangle with vertices P at negative 3 comma 3, Q at negative 3 comma 9, and R at negative 12 comma 9.

What sequence of transformations proves ΔMNO ~ ΔPQR?

ΔMNO was dilated by a factor of 3 about the origin, then reflected across the x-axis to form ΔPQR.
ΔMNO was dilated by a factor of 3 about from the origin, then reflected across the y-axis to form ΔPQR.
ΔMNO was dilated by a factor of 4 about the origin, then rotated 180° clockwise about the origin to form ΔPQR.
ΔMNO was dilated by a factor of 4 about the origin, then translated left 7 units to form ΔPQR.

Answer 1

The correct sequence of transformations involves dilating triangle MNO by a factor of 3 about the origin, and then reflecting the resulting triangle across the y-axis to form triangle PQR.

Step-by-step explanation:

The question involves identifying the sequence of transformations that prove that triangles MNO and PQR are similar on the coordinate plane. To determine the correct sequence of transformations, we look at the coordinates of the vertices of both triangles. Triangle MNO has vertices at M(1, 1), N(1, 3), and O(4, 3), and triangle PQR has vertices at P(-3, 3), Q(-3, 9), and R(-12, 9). By examining the proportional relationships between the sides of the two triangles, we can determine the scale factor of the dilation. The distance between N and O in ∆MNO is 3 units, and the distance between Q and R in ∆PQR is 9 units, which is 3 times larger. Therefore, the scale factor for the dilation is 3.

Next, we consider the directions of the coordinates. To transform ∆MNO into ∆PQR after dilation by a scale factor of 3, we must reflect across an axis. Since the x-coordinates of ∆PQR are negative and the y-coordinates have remained positive, the reflection must have been across the y-axis. The correct sequence of transformations to prove that ∆MNO is similar to ∆PQR is that ∆MNO was dilated by a factor of 3 about the origin, then reflected across the y-axis to form ∆PQR. Therefore, the correct option is: ∆MNO was dilated by a factor of 3 about from the origin, then reflected across the y-axis to form ∆PQR.