Question

a rectangle is dilated by a scale factor of 0.5 with the origin as the center of dilation. what happens to the coordinates of its vertices?
a. they double.
b. they become negative.
c. they are halved.
d. they remain unchanged.

two triangles have corresponding sides with ratios of \\( \frac { 3 } { 2 } \\), \\( \frac { 6 } { 4 } \\), and \\( \frac { 9 } { 6 } \\). what does this indicate about the two triangles?
a. they cannot be related.
b. they are reflections of one another.
c. they are similar.
d. they are congruent.

a triangle has vertices \\( a ( 1,2 ) \\), \\( b ( 3,4 ) \\), and \\( c ( 2,6 ) \\). after a transformation, the vertices become \\( a ^ { prime } ( 4,5 ) \\), \\( b ^ { prime } ( 6,7 ) \\), and \\( c ^ { prime } ( 5,9 ) \\). what is the translation rule?
a. \\( ( x + 3, y + 3 ) \\)
b. \\( ( x + 3, y - 3 ) \\)
c. \\( ( x + 2, y + 3 ) \\)
d. \\( ( x - 3, y + 2 ) \\)

if two figures are similar, which of the following must be true?
a. they have the same area.
b. they are the same size.
c. they are congruent.
d. their corresponding angles are equal.

if two triangles have corresponding angles that are equal, what can be said about the triangles?
a. they have different shapes.
b. they are reflections.
c. they are similar.
d. they are congruent.

what is the main characteristic of a dilation transformation?
a. it rotates the figure.
b. it changes the size of the figure.
c. it slides the figure.
d. it flips the figure.

Explanation:

First question:

Step1: Understand dilation

Dilation with scale factor \(k = 0.5=\frac{1}{2}\). If a point \((x,y)\) is dilated about the origin, the new point is \((kx,ky)\). So \((x,y)\to(0.5x,0.5y)\), which means the coordinates are halved.

Second question:

Step1: Recall similarity condition

For two triangles, if the ratios of their corresponding sides are equal (\(\frac{a}{a'}=\frac{b}{b'}=\frac{c}{c'}\)), by the Side - Side - Side (SSS) similarity criterion, the triangles are similar. Here \(\frac{3}{6}=\frac{4}{8}=\frac{5}{10}=\frac{1}{2}\).

Third question:

Step1: Calculate \(x\) - coordinate change

For point \(A(1,2)\) to \(A'(4,5)\), \(x\) - coordinate: \(4 - 1=3\). For point \(B(3,4)\) to \(B'(6,7)\), \(x\) - coordinate: \(6 - 3 = 3\). For point \(C(2,6)\) to \(C'(5,9)\), \(x\) - coordinate: \(5 - 2=3\).

Step2: Calculate \(y\) - coordinate change

For point \(A(1,2)\) to \(A'(4,5)\), \(y\) - coordinate: \(5 - 2=3\). For point \(B(3,4)\) to \(B'(6,7)\), \(y\) - coordinate: \(7 - 4 = 3\). For point \(C(2,6)\) to \(C'(5,9)\), \(y\) - coordinate: \(9 - 6=3\). The translation rule is \((x + 3,y+3)\).

Fourth question:

Step1: Recall properties of similar figures

Similar figures have the same shape (corresponding angles are equal), but not necessarily the same size (so area and side - lengths can be different). Congruent figures are a special case of similar figures with scale factor \(k = 1\).

Fifth question:

Step1: Recall triangle similarity criteria

If two triangles have corresponding angles equal (by AA - Angle - Angle similarity criterion), the triangles are similar.

Sixth question:

Step1: Recall dilation definition

Dilation is a transformation that changes the size of a figure. It does not rotate (that's rotation), does not slide (that's translation), and does not flip (that's reflection).

Answer:

1. c. They are halved.
2. c. They are similar.
3. a. \((x + 3,y + 3)\)
4. d. Their corresponding angles are equal.
5. c. They are similar.
6. b. It changes the size of the figure.