Question

triangle \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) is the result of dilating \\( \triangle a b c \\) about point \\( b \\) by a scale factor of \\( \frac { 4 } { 3 } \\). determine whether each claim about the properties of \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) is true or false. the measures of \\( \angle c \\) and \\( \angle c ^ { prime } \\) are equal the coordinates of \\( c \\) and \\( c ^ { prime } \\) are the same

Explanation:

Step1: Properties of dilation

Dilation is a transformation that changes the size of a figure but not its shape. Corresponding angles of the original figure and the dilated figure are equal. So, for \(\triangle ABC\) and \(\triangle A'B'C'\) (where \(\triangle A'B'C'\) is a dilation of \(\triangle ABC\)), \(\angle C\) and \(\angle C'\) are corresponding angles.
Since dilation preserves angle - measures, \(\angle C\cong\angle C'\), so the measure of \(\angle C\) and \(\angle C'\) are equal.

Step2: Coordinates after dilation

The formula for dilation about a point \(B(x_0,y_0)\) with a scale factor \(k\) for a point \(C(x,y)\) is \(C'(x_0 + k(x - x_0),y_0 + k(y - y_0))\). Here \(B(- 3,0)\), \(C(2,3)\) and \(k=\frac{4}{3}\)

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The coordinates of \(C\) are \((2,3)\) and the coordinates of \(C'\) are \((\frac{11}{3},4)\) (using the dilation formula \( (x',y')=(x_B + k(x - x_B),y_B + k(y - y_B))\) where \((x_B,y_B)=(-3,0)\), \((x,y)=(2,3)\) and \(k = \frac{4}{3}\)). So the coordinates of \(C\) and \(C'\) are not the same.

Answer:

The measure of \(\angle C\) and \(\angle C'\) are equal: True.
The coordinates of \(C\) and \(C'\) are the same: False.