Question

question: 7
select all the true statements.
a. dilations always increase the length of line segments.
b. dilations take perpendicular lines to perpendicular lines.
c. dilations of an angle are congruent to the original angle.
d. dilations increase the measure of angles.
e. dilations of a triangle are congruent to the original triangle.
f. dilations of a triangle are similar to the origi triangle.

Explanation:

Step1: Analyze option A

If the scale factor of a dilation is between \(0\) and \(1\), the length of line segments decreases. So, dilations do not always increase the length of line segments.

Step2: Analyze option B

Dilations preserve the angle between lines. If two lines are perpendicular (\(90^{\circ}\) angle between them), after dilation, the angle between their images is still \(90^{\circ}\).

Step3: Analyze option C

Dilations preserve the measure of angles. So, dilations of an angle are congruent to the original angle.

Step4: Analyze option D

As mentioned in step 3, dilations do not change the measure of angles.

Step5: Analyze option E

Dilations change the size of a triangle (unless the scale factor is \(1\)). Congruent triangles have the same size and shape. So, dilations of a triangle are not congruent to the original triangle (except when scale factor \(k = 1\)).

Step6: Analyze option F

By the definition of similar figures (same shape, proportional sides), dilations of a triangle are similar to the original triangle.

Answer:

B. Dilations take perpendicular lines to perpendicular lines, C. Dilations of an angle are congruent to the original angle, F. Dilations of a triangle are similar to the original triangle.