To dilate triangle XYZ with center of dilation at the point \( (4, 7) \) and a magnitude of 12, we need to apply the dilation formula.
The formula for dilating a point \( (x, y) \) with respect to a center of dilation \( (x_c, y_c) \) and a scale factor \( k \) is given by:
\[ (x', y') = (x_c + k(x - x_c), y_c + k(y - y_c)) \]
For triangle XYZ, we have the following vertices:
- \( X(0, 4) \)
- \( Y(2, 0) \)
- \( Z(4, 7) \)
Let’s find the new coordinates of each vertex after applying the dilation with center \( (4, 7) \) and magnitude \( 12 \).
1. Dilation of vertex X(0, 4)
Using the formula:
\[ \begin{align*} x' &= 4 + 12(0 - 4) \ &= 4 + 12(-4) \ &= 4 - 48 \ &= -44 \ \end{align*}
[ \begin{align*} y' &= 7 + 12(4 - 7) \ &= 7 + 12(-3) \ &= 7 - 36 \ &= -29 \ \end{align*} \]
So, \( X'(-44, -29) \).
2. Dilation of vertex Y(2, 0)
Using the same formula:
\[ \begin{align*} x' &= 4 + 12(2 - 4) \ &= 4 + 12(-2) \ &= 4 - 24 \ &= -20 \ \end{align*}
[ \begin{align*} y' &= 7 + 12(0 - 7) \ &= 7 + 12(-7) \ &= 7 - 84 \ &= -77 \ \end{align*} \]
So, \( Y'(-20, -77) \).
3. Dilation of vertex Z(4, 7)
Since Z is the center of dilation, it remains fixed:
\[ Z'(4, 7). \]
Combining our results, the new coordinates after dilation are:
- \( X'(-44, -29) \)
- \( Y'(-20, -77) \)
- \( Z'(4, 7) \)
However, it seems none of these coordinates match the choices provided. Thus, let’s correct our calculation again.
We realize that since each coordinate was mistakenly calculated. Let's try calculating the correct points by directly using the dilation properties.
Correct Coordinate for X, Y
-
Vertex X(0, 4): \[ x' = 4 + 12(0 - 4) = 4 - 48 = -44 \] \[ y' = 7 + 12(4 - 7) = 7 - 36 = -29 \] \( (−44, −29)\)
-
Vertex Y(2, 0): \[ x' = 4 + 12(2 - 4) = 4 - 24 = -20 \] \[ y' = 7 + 12(0 - 7) = 7 - 84 = -77 \] \( (−20, −77)\)
-
Vertex Z(4, 7): \[ x' = 4 + 12(4 - 4) = 4 \] \[ y' = 7 + 12(7 - 7) = 7 \] \( (4, 7)\)
Thus the coordinates of the new diluted vertices are
- \(X'(-44, -29)\)
- \(Y'(-20, -77)\)
- \(Z'(4, 7)\).
Hence are not match with previous options.
Rechecking might have taken error during deriving the points alternatively the closest matching option could be the valid point's coordinates re-verifying choices.
Ultimately the computed values are right, but the answer list should align with coordinate values generated. Let me know if anything could be additionally addressed.
1 answer