Question

1. quadrilateral efgh is a scaled - copy of quadrilateral abcd. select all the true statements.

a. the segment ef has twice the length of segment ab.
b. the segment cd has twice the length of segment fg.
c. the measure of angle hef is twice the measure of angle dab.
d. the length of segment eh is 16 units.
e. the area of efgh is twice the area of abcd.

Explanation:

Step1: Find the scale - factor

We know that if one figure is a scaled - copy of another, the ratio of corresponding side lengths is the same. Given \(CD = 6\) and \(GH=12\), the scale - factor \(k=\frac{GH}{CD}=\frac{12}{6} = 2\).

Step2: Analyze option A

Corresponding sides of similar figures are in proportion. If \(EF\) corresponds to \(AB\), and the scale - factor \(k = 2\), then \(EF=2AB\). Since \(AB = 4\), \(EF = 8\). Option A is correct.

Step3: Analyze option B

\(CD = 6\) and \(FG\) corresponds to \(BC\). Since \(BC = 3\) and the scale - factor \(k = 2\), \(FG=2BC = 6\). So \(CD=FG\), and option B is incorrect.

Step4: Analyze option C

In similar figures, corresponding angles are equal. So \(\angle HEF=\angle DAB\), not \(\angle HEF = 2\angle DAB\). Option C is incorrect.

Step5: Analyze option D

If \(EH\) corresponds to \(AD\) and \(AD = 8\), with a scale - factor \(k = 2\), then \(EH=2AD=16\). Option D is correct.

Step6: Analyze option E

If the scale - factor of two similar figures is \(k\), the ratio of their areas is \(k^{2}\). Here \(k = 2\), so the area of \(EFGH\) is \(k^{2}=4\) times the area of \(ABCD\), not \(2\) times. Option E is incorrect.

Answer:

A. The segment \(EF\) has twice the length of the segment \(AB\).
D. The length of the segment \(EH\) is 16 units.