Question

2 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 original triangle.

Explanation:

Step1: Analyze option A

If the scale factor of dilation \(k<1\), the length of line segments will decrease. For example, if \(k = 0.5\), a line segment of length \(l\) becomes \(0.5l\). So, option A is false.

Step2: Analyze option B

Dilation is a similarity transformation. If two lines are perpendicular (\(90^{\circ}\) angle between them), after dilation, the angle between their images is still \(90^{\circ}\). Because dilation preserves the measure of angles. So, option B is true.

Step3: Analyze option C

By the property of dilation, the measure of an angle is preserved. Let \(\angle ABC\) be an angle. After dilation with scale factor \(k\), the image \(\angle A'B'C'\) has \(m\angle ABC=m\angle A'B'C'\). So, option C is true.

Step4: Analyze option D

Since dilation preserves the measure of angles (as shown in step 3), it does not increase the measure of angles. So, option D is false.

Step5: Analyze option E

For dilation with scale factor \(k
eq1\), the sides of the triangle change. If \(\triangle ABC\) is dilated with \(k = 2\), \(AB
eq A'B'\), \(BC
eq B'C'\), \(AC
eq A'C'\). Congruent triangles require \(AB = A'B'\), \(BC=B'C'\), \(AC = A'C'\) (by SSS criterion). So, option E is false.

Step6: Analyze option F

Dilation is a similarity transformation. If \(\triangle ABC\) is dilated to \(\triangle A'B'C'\) with scale factor \(k\), then \(\frac{AB}{A'B'}=\frac{BC}{B'C'}=\frac{AC}{A'C'}=k\) and the corresponding angles are equal (\(m\angle A=m\angle A'\), \(m\angle B=m\angle B'\), \(m\angle C=m\angle C'\)). By the SSS (side - side - side) similarity criterion, \(\triangle ABC\sim\triangle A'B'C'\). So, option F is true.

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.