Answer :
To determine if trapezoid ABDC is the result of a dilation of trapezoid MNPQ by a scale factor of [tex]\(\frac{2}{5}\)[/tex], we need to check if both pairs of corresponding sides in the trapezoids are proportional by the given scale factor.
A dilation requires that each corresponding pair of sides between the two shapes is proportional by the same scale factor. In this case, the scale factor is [tex]\(\frac{2}{5}\)[/tex].
1. Identify the corresponding sides of the trapezoids:
- [tex]\(AB\)[/tex] corresponds to [tex]\(MN\)[/tex]
- [tex]\(CD\)[/tex] corresponds to [tex]\(QP\)[/tex]
2. Check the given conditions:
- The problem states that [tex]\(AB\)[/tex] is [tex]\(\frac{2}{5}\)[/tex] the length of [tex]\(MN\)[/tex].
- The problem also states that [tex]\(CD\)[/tex] is [tex]\(\frac{1}{3}\)[/tex] the length of [tex]\(QP\)[/tex].
3. Analyze the conditions:
- For a correct dilation with a scale factor of [tex]\(\frac{2}{5}\)[/tex], [tex]\(CD\)[/tex] should also be [tex]\(\frac{2}{5}\)[/tex] the length of [tex]\(QP\)[/tex].
- Since [tex]\(CD\)[/tex] is not [tex]\(\frac{2}{5}\)[/tex] but [tex]\(\frac{1}{3}\)[/tex] of [tex]\(QP\)[/tex], the condition for a dilation is not fully met.
Therefore, the trapezoid ABDC is not a result of the dilation of trapezoid MNPQ by a scale factor of [tex]\(\frac{2}{5}\)[/tex] because although [tex]\(AB\)[/tex] is proportional to [tex]\(MN\)[/tex] by that scale factor, [tex]\(CD\)[/tex] does not match the proportionality requirement with [tex]\(QP\)[/tex]. The correct answer is:
No, because AB is [tex]\(\frac{2}{5}\)[/tex] the length of MN but CD is [tex]\(\frac{1}{3}\)[/tex] the length QP.
A dilation requires that each corresponding pair of sides between the two shapes is proportional by the same scale factor. In this case, the scale factor is [tex]\(\frac{2}{5}\)[/tex].
1. Identify the corresponding sides of the trapezoids:
- [tex]\(AB\)[/tex] corresponds to [tex]\(MN\)[/tex]
- [tex]\(CD\)[/tex] corresponds to [tex]\(QP\)[/tex]
2. Check the given conditions:
- The problem states that [tex]\(AB\)[/tex] is [tex]\(\frac{2}{5}\)[/tex] the length of [tex]\(MN\)[/tex].
- The problem also states that [tex]\(CD\)[/tex] is [tex]\(\frac{1}{3}\)[/tex] the length of [tex]\(QP\)[/tex].
3. Analyze the conditions:
- For a correct dilation with a scale factor of [tex]\(\frac{2}{5}\)[/tex], [tex]\(CD\)[/tex] should also be [tex]\(\frac{2}{5}\)[/tex] the length of [tex]\(QP\)[/tex].
- Since [tex]\(CD\)[/tex] is not [tex]\(\frac{2}{5}\)[/tex] but [tex]\(\frac{1}{3}\)[/tex] of [tex]\(QP\)[/tex], the condition for a dilation is not fully met.
Therefore, the trapezoid ABDC is not a result of the dilation of trapezoid MNPQ by a scale factor of [tex]\(\frac{2}{5}\)[/tex] because although [tex]\(AB\)[/tex] is proportional to [tex]\(MN\)[/tex] by that scale factor, [tex]\(CD\)[/tex] does not match the proportionality requirement with [tex]\(QP\)[/tex]. The correct answer is:
No, because AB is [tex]\(\frac{2}{5}\)[/tex] the length of MN but CD is [tex]\(\frac{1}{3}\)[/tex] the length QP.
