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

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

The perimeters of $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ are the same.
True/false

The lengths of $\overline{C D}$ and $\overline{C^{\prime} D^{\prime}}$ are equal.
True/false

Question

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

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

The perimeters of $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ are the same.
True/false

The lengths of $\overline{C D}$ and $\overline{C^{\prime} D^{\prime}}$ are equal.
True/false

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the truth of the first statement, we need to calculate the perimeter of the original quadrilateral ABCD and compare it to the perimeter of the dilated quadrilateral A'B'C'D'. The dilation factor is $\frac{3}{2}$, which means that all sides of the quadrilateral will be scaled by this factor. ‖ ‖ 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 $sqrt(17)$, $4$, $sqrt(61)$, and $2$. ‖ ‖ step 3 ⋮ The perimeter of ABCD is $sqrt(17) + 4 + sqrt(61) + 2$. ‖ ‖ step 4 ⋮ To find the perimeter of the dilated quadrilateral A'B'C'D', we multiply each side of ABCD by the scale factor $\frac{3}{2}$. Thus, the perimeter of A'B'C'D' is $\frac{3}{2}(sqrt(17) + 4 + sqrt(61) + 2)$. ‖ ‖ step 5 ⋮ Simplifying the expression for the perimeter of A'B'C'D', we get $\frac{3}{2} imes (sqrt(17) + 4 + sqrt(61) + 2) eq sqrt(17) + 4 + sqrt(61) + 2$. 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 $sqrt(61)$. ‖ ‖ step 7 ⋮ The length of C'D' after dilation is $\frac{3}{2} imes sqrt(61)$, which is not equal to $sqrt(61)$. Therefore, the lengths of CD and C'D' are not equal. ‖ ∻[question 1] Answer∻ ***False*** ∻[question 2] Answer∻ ***False*** ∻Key Concept∻ ⚹Dilation and Perimeter⚹ ∻Explanation∻ ⚹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⚹ ∻Explanation∻ ⚹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.⚹◊What is the formula to calculate the perimeter of a quadrilateral on a coordinate plane?⍭ Generate me a similar question◊

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