Question

1. (03.01 mc)

\overline{xy} has endpoints located at x (0, 8) and y (0, 2). it was dilated at a scale factor of 2 from center (0, 2). which statement describes the pre - image? (1 point)
\overline{xy} is located at x (0, 1) and y (0, 5) and is half the length of \overline{xy}.
\overline{xy} is located at x (0, 1) and y (0, 5) and is twice the length of \overline{xy}.
\overline{xy} is located at x (0, 5) and y (0, 2) and is half the length of \overline{xy}.
\overline{xy} is located at x (0, 5) and y (0, 2) and is twice the length of \overline{xy}.

1. (03.01 lc)

grid with points a and b

Explanation:

Step1: Calculate the length of \(\overline{X'Y'}\)

The distance formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\vert y_2 - y_1\vert\) (since \(x_1=x_2 = 0\) for \(X'(0,8)\) and \(Y'(0,2)\)). So, \(d_{X'Y'}=\vert8 - 2\vert=6\).

Step2: Use the dilation formula

If a segment is dilated with scale factor \(k = 2\) from a center, and let the length of the pre - image be \(d\). Then \(d_{X'Y'}=k\times d\). Since \(k = 2\) and \(d_{X'Y'}=6\), we can solve for \(d\) (the length of the pre - image \(\overline{XY}\)): \(d=\frac{d_{X'Y'}}{k}\). Substituting \(k = 2\) and \(d_{X'Y'}=6\), we get \(d = 3\).

Step3: Find the endpoints of the pre - image

Let the center of dilation be \(C(0,2)\). For a dilation with scale factor \(k = 2\), if a point \(P'(x',y')\) is the image of a point \(P(x,y)\) with center of dilation \(C(x_c,y_c)\), the formula is \((x',y')=(x_c+(x - x_c)\times k,y_c+(y - y_c)\times k)\). For \(X'(0,8)\) and \(C(0,2)\), let \(X(x,y)\) be the pre - image. Then \(8=2+(y - 2)\times2\). Solving for \(y\):

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For \(Y'(0,2)\), since \(Y'(0,2)\) and \(C(0,2)\), the pre - image \(Y\) has \(y = 2\) (because \((x',y')=(x_c+(x - x_c)\times k,y_c+(y - y_c)\times k)\), when \(y'=y_c\), \(y=y_c\)).

Answer:

C. \(\overline{XY}\) is located at \(X(0,5)\) and \(Y(0,2)\) and is half the length of \(\overline{X'Y'}\)