Question

1. trapezoid dilation properties 6a. trapezoid wxyz is dilated with scale factor 3. which is true? a. △wxy ~ △wxy b. side lengths are tripled. c. angle measures are tripled. d. perimeter stays the same. 6b. trapezoid jklm is dilated by factor ½. which is true? there are two right answers. a. the perimeter is halved. b. the side lengths are halved. c. the angles are halved. d. the area is halved. 6c. a trapezoid is enlarged with scale factor 4. which is true? a. side lengths are multiplied by 4. b. angles are multiplied by 4. c. the perimeter remains unchanged. d. the figure is congruent. 6d. a trapezoid is reduced with scale factor ¼. which is true? a. side lengths are multiplied by ¼. b. angles are reduced by ¼. c. area is reduced by ¼. d. figure becomes congruent.

Explanation:

Step1: Recall dilation properties

When a figure is dilated by a scale - factor \(k\), the side - lengths are multiplied by \(k\), the perimeter is multiplied by \(k\), the angles remain the same, and the area is multiplied by \(k^{2}\).

6a. Trapezoid \(WXYZ\) is dilated with a scale factor

If the scale factor is \(k\), side - lengths are multiplied by \(k\). Here, if \(k = 3\) (tripled), side lengths are tripled. Angle measures remain the same in a dilation, and the perimeter is also multiplied by the scale factor. The area is multiplied by \(k^{2}=9\). So the correct statements are:

• B. Side lengths are tripled.
• C. Angle measures are tripled (False, angle measures stay the same).
• D. Perimeter stays the same (False, perimeter is multiplied by the scale factor).

The correct answer for 6a is B.

6b. Trapezoid \(JKLM\) is dilated by factor \(\frac{1}{2}\)

When \(k=\frac{1}{2}\), side - lengths are multiplied by \(\frac{1}{2}\) (halved), the perimeter is multiplied by \(\frac{1}{2}\) (halved), the angles remain the same, and the area is multiplied by \(k^{2}=\frac{1}{4}\) (quartered).

• A. The perimeter is halved (True, since perimeter \(P\) of a polygon with side - lengths \(s_1,s_2,\cdots\) is \(P = s_1 + s_2+\cdots\), and if each \(s_i\) is multiplied by \(\frac{1}{2}\), \(P\) is multiplied by \(\frac{1}{2}\)).
• B. The side lengths are halved (True, as side - lengths are multiplied by the scale factor \(\frac{1}{2}\)).
• C. The angles are halved (False, angles remain the same in dilation).
• D. The area is halved (False, area is multiplied by \(k^{2}=\frac{1}{4}\)).

The correct answers for 6b are A and B.

6c. A trapezoid is enlarged with scale factor 4

When \(k = 4\), side - lengths are multiplied by 4, the perimeter is multiplied by 4, the angles remain the same, and the area is multiplied by \(k^{2}=16\).

• A. Side lengths are multiplied by 4 (True).
• B. Angles are multiplied by 4 (False, angles remain the same).
• C. The perimeter remains unchanged (False, perimeter is multiplied by 4).
• D. The figure is congruent (False, congruent figures have a scale factor of 1).

The correct answer for 6c is A.

6d. A trapezoid is reduced with scale factor \(\frac{1}{3}\)

When \(k=\frac{1}{3}\), side - lengths are multiplied by \(\frac{1}{3}\), the perimeter is multiplied by \(\frac{1}{3}\), the angles remain the same, and the area is multiplied by \(k^{2}=\frac{1}{9}\).

• A. Side lengths are multiplied by \(\frac{1}{3}\) (True).
• B. Angles are reduced by \(\frac{1}{3}\) (False, angles remain the same).
• C. Area is reduced by \(\frac{1}{3}\) (False, area is multiplied by \(k^{2}=\frac{1}{9}\)).
• D. Figure becomes congruent (False, congruent figures have a scale factor of 1).

The correct answer for 6d is A.

Answer:

6a. B
6b. A, B
6c. A
6d. A