Dilation the figure using the indicated scale factor $k$ . What is the value of the ratio (new to original perimeters? The areas? a triangle with vertices $(0, 0), (0, 2)$ and $(2, 0); k = 3$

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Solution

Step 1: Find the coordinates of image after dilation.
Given, coordinates are $(0, 0), (0, 2), (2, 0)$ and scale factor, $k = 3$ .
As we know after dilation the coordinates changes as,
$P (x, y) = P' (x \times a, y \times a)$
Using the above criteria of dilation with scale factor, $k = 3$
$A (0, 0) = A' (0 \times 3, 0 \times 3) = A' (0, 0) B (0, 2) = B' (0 \times 3, 2 \times 3) = B' (0, 6) C (2, 0), C' (2 \times 3, 0 \times 3) = C' (6, 0)$
Step 2: Find the length of sides of triangle.
After dilation, coordinates are $A' (0, 0), B' (0, 6), C' (6, 0)$
As we know length of side AB having coordinates $A (x_{1}, y_{1)}$ and $B (x_{2}, y_{2})$ :
$= \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$
So, the length of line AB = $\sqrt{{(0 - 0)}^{2} + {(2 - 0)}^{2}} = 2$
And, the length of line A'B'= $\sqrt{{(0 - 0)}^{2} + {(6 - 0)}^{2}} = 6$
Step 3: Find the ratio of perimeter of new triangle to the original triangle.
Since, we know that when two figures are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
$\Rightarrow \frac{P e r i m e t e r o f n e w t r i a n g l e}{P e r i m e t e r o f o r i g i n a l t r i a n g l e} = \frac{S i d e l e n g t h o f n e w t r i a n g l e}{S i d e l e n g t h o f o r i g i n a l t r i a n g l e}$
So, the ratio of perimeter of new triangle to the original triangle is
$\Rightarrow \frac{P e r i m e t e r o f n e w t r i a n g l e}{P e r i m e t e r o f o r i g i n a l t r i a n g l e} = \frac{6}{2} = 3$
Step 4: Find the ratio of area of new triangle to the original triangle.
Since, we know that when two figures are similar then the value of the ratio of their area is equal to the value of the ratio of their corresponding square of side lengths.
$\Rightarrow \frac{A r e a o f n e w t r i a n g l e}{A r e a o f o r i g i n a l t r i a n g l e} = \frac{{(S i d e l e n g t h o f n e w t r i a n g l e)}^{2}}{{(S i d e l e n g t h o f o r i g i n a l t r i a n g l e)}^{2}}$
So, the ratio of area of new triangle to the original triangle is
$\Rightarrow \frac{A r e a o f n e w t r i a n g l e}{A r e a o f o r i g i n a l t r i a n g l e} = \frac{6^{2}}{2^{2}} = 9$
Step 5: Final answer.
Hence, the ratio of perimeter and area of new triangle to the original triangle is $3$ and $9$ respectively.

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Dilation the figure using the indicated scale factor k. What is the value of the ratio (new to original perimeters? The areas? a triangle with vertices (0,0),(0,2) and (2,0);k=3

Dilation the figure using the indicated scale factor $k$ . What is the value of the ratio (new to original perimeters? The areas? a triangle with vertices $(0, 0), (0, 2)$ and $(2, 0); k = 3$