Question

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This diagram shows a pre-image
△ABC, and its image, A′′B′′C′′, after a series of transformations.
Select from the drop-down menus to correctly complete the statements.

Answer 1

△ABC is rotated 180 degrees about the origin to become △A'B'C'. Then △A'B'C' is translated 3 units down to become △A"B"C". Because the transformations are both rigid, the pre-image and image are congruent.

In Euclidean Geometry, the mapping rule for the rotation of a geometric figure about the origin by 180° in a clockwise or counterclockwise direction can be modeled by the following mathematical expression:

(x, y) → (-x, -y)

Point A (1, -1) → Point A' (-1, 1)

Point B (4, -2) → Point B' (-4, 2)

Point C (7, 2) → Point C' (-7, -2)

Next, we would apply a translation 3 units down to the new figure (△A'B'C'), in order to determine the coordinates of its image as follows;

(x, y) → (x, y - 3)

A' (-1, 1) → (-1, 1 - 3) = A" (-1, -2).

B' (-4, 2) → (-4, 2 - 3) = B" (-4, -1).

C' (-7, -2) → (-7, -2 - 3) = C" (-7, -5).

Therefore, the resulting figure must be located in the third quadrant as illustrated in the diagram shown above.

Answer 2

Explanation:

Given that triangle ABC is transformed two times to get triangle

A"B"C"

First ABC is transformed in to A'B'C' as follows:

The coordinates of A are (1,-1) and transformed into A'(-1,1)

Similarly B (4,-2) became B'(-4,2) and

C(7,-2) became C'(-7,2)

i.e. (x,y) becomes (-x,-y)

This is nothing but reflection about a point here origin.

Thus first transformation is reflection on the origin.

Next is exactly shifting 3 units down