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Posted 2 months ago

Quadrilateral ABCDA^{\prime} B^{\prime} C^{\prime} D^{\prime} is the result of dilating quadrilateral ABCDA B C D about point DD by a scale factor of 32\frac{3}{2}.

Determine whether each claim about the properties of ABCDA B C D and ABCDA^{\prime} B^{\prime} C^{\prime} D^{\prime} is true or false.

The perimeters of ABCDA B C D and ABCDA^{\prime} B^{\prime} C^{\prime} D^{\prime} are the same.

The lengths of CD\overline{C D} and CD\overline{C^{\prime} D^{\prime}} are equal.
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 2
The perimeter of the original quadrilateral ABCD can be found by adding the distances between consecutive vertices. From the asksia-ll calculator, we have the distances as sqrt(17)sqrt(17), 44, sqrt(61)sqrt(61), and 22
step 3
The perimeter of ABCD is sqrt(17)+4+sqrt(61)+2sqrt(17) + 4 + sqrt(61) + 2
step 4
To find the perimeter of the dilated quadrilateral A'B'C'D', we multiply each side of ABCD by the scale factor 32\frac{3}{2}. Thus, the perimeter of A'B'C'D' is 32(sqrt(17)+4+sqrt(61)+2)\frac{3}{2}(sqrt(17) + 4 + sqrt(61) + 2)
step 5
Simplifying the expression for the perimeter of A'B'C'D', we get 32×(sqrt(17)+4+sqrt(61)+2)sqrt(17)+4+sqrt(61)+2\frac{3}{2} \times (sqrt(17) + 4 + sqrt(61) + 2) \neq sqrt(17) + 4 + sqrt(61) + 2. Therefore, the perimeters of ABCD and A'B'C'D' are not the same
step 6
For the second statement, the length of CD is the distance between points C and D, which is sqrt(61)sqrt(61)
step 7
The length of C'D' after dilation is 32×sqrt(61)\frac{3}{2} \times sqrt(61), which is not equal to sqrt(61)sqrt(61). Therefore, the lengths of CD and C'D' are not equal
[question 1] Answer
[question 2] Answer
Key Concept
Dilation and Perimeter
When a figure is dilated by a scale factor, the lengths of the sides are multiplied by that factor. This changes the perimeter proportionally to the scale factor.
Key Concept
Dilation and Side Lengths
The length of a side after dilation is the original length multiplied by the scale factor. If the scale factor is not 1, the dilated length is different from the original length.

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