QuestionA triangle transforms from $XY Z$ (perimeter 40) to $X'Y'Z'$ (perimeter 100). Find the transformation and $XY, Y'Z'$ . Options: a, b, c, d.

Studdy Solution

STEP 1

Assumptions1. The perimeter of triangle $XYZ$ is40. . The perimeter of triangle $X'Y'Z'$ is100.
3. The length of side $X'Z'$ is20.
4. The length of side $YZ$ is15.
5. We are looking for the type of transformation and the lengths of sides $XY$ and $Y'Z'$ .

STEP 2

First, we need to find the scale factor of the transformation. We can do this by dividing the perimeter of triangle $X'Y'Z'$ by the perimeter of triangle $XYZ$ .
$Scale\, factor = \frac{Perimeter\, of\, triangle\, X'Y'Z'}{Perimeter\, of\, triangle\, XYZ}$

STEP 3

Now, plug in the given values for the perimeters of the triangles to calculate the scale factor.
$Scale\, factor = \frac{100}{40}$

STEP 4

Calculate the scale factor.
$Scale\, factor = \frac{100}{40} =2.$

STEP 5

Now that we have the scale factor, we can determine the type of transformation. A scale factor of2.5 indicates a dilation, not a stretch or translation. So, the transformation is a dilation.

STEP 6

Next, we need to find the length of side $XY$ . We can do this by subtracting the length of side $YZ$ from the perimeter of triangle $XYZ$ and then dividing by2 (since $XY$ and $XZ$ are the remaining sides and they are equal in length).
$XY = \frac{Perimeter\, of\, triangle\, XYZ - YZ}{2}$

STEP 7

Plug in the values for the perimeter of triangle $XYZ$ and the length of side $YZ$ to calculate the length of side $XY$ .
$XY = \frac{40 -15}{2}$

STEP 8

Calculate the length of side $XY$ .
$XY = \frac{40 -15}{2} =12.5$

STEP 9

Now that we have the length of side $XY$ , we can find the length of side $Y'Z'$ . We can do this by subtracting the lengths of sides $X'Y'$ and $X'Z'$ from the perimeter of triangle $X'Y'Z'$ (since $X'Y' = XY$ and $X'Z' = XZ$ ).
$Y'Z' = Perimeter\, of\, triangle\, X'Y'Z' - X'Y' - X'Z'$

STEP 10

Plug in the values for the perimeter of triangle $X'Y'Z'$ , and the lengths of sides $X'Y'$ and $X'Z'$ to calculate the length of side $Y'Z'$ .
$Y'Z' =100 -12.5 -20$

STEP 11

Calculate the length of side $Y'Z'$ .
$Y'Z' =100 -.5 -20 =67.5$ Therefore, the transformation must have been a dilation with a perimeter scale factor of.5, and the lengths of sides $XY$ and $Y'Z'$ are.5 and67.5, respectively. The correct answer is not listed among the options provided.

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