Question

5 from unit 2, lesson 3 a rotation by angle ace using point c as the center takes triangle cba onto triangle cde. a. explain why the image of segment cb lines up with segment cd. b. explain why the image of b coincides with d. c. is triangle abc congruent to triangle edc? explain your reasoning.

Explanation:

Step1: Rotation property

A rotation is a transformation that preserves the length of segments and the measure of angles. When we rotate around point \(C\) by angle \(ACE\), the angle between \(CB\) and \(CD\) is equal to the rotation angle. Since \(CB\) is rotated by angle \(ACE\) (which is the angle between \(CB\) and \(CD\) around point \(C\)), the image of segment \(CB\) lines up with segment \(CD\) because rotation about a point \(C\) moves a segment \(CB\) such that the angle of rotation is the angle between the original and the image segment at the center of rotation.

Step2: Congruent segments

In a rotation, the distance from the center of rotation to a point is preserved. So \(CB = CD\) (because a rotation about \(C\) maps \(B\) to a point \(D\) such that \(CB\) is rotated to \(CD\) and rotation preserves distances from the center of rotation. Since \(CB\) is rotated to \(CD\) (from part a) and \(CB=CD\), the image of \(B\) (which is on the circle centered at \(C\) with radius \(CB\)) coincides with \(D\) (which is on the same - radius circle centered at \(C\))

Step3: Congruence of triangles

Triangles \(ABC\) and \(EDC\) are congruent. A rotation is a rigid transformation. Rigid transformations (such as rotation) preserve side - lengths and angle - measures. So, if a rotation maps \(\triangle CBA\) to \(\triangle CDE\), then \(AB = ED\), \(BC = DC\), \(AC=EC\) (by the property of rotation: \(r(C,\theta)(X)\) implies \(CX = C r(C,\theta)(X)\) for any point \(X\) and rotation \(r(C,\theta)\) about point \(C\) by angle \(\theta\)), and \(\angle ABC=\angle EDC\), \(\angle BAC=\angle DEC\), \(\angle ACB=\angle ECD\). By the Side - Side - Side (SSS) criterion (or Side - Angle - Side (SAS) criterion, since we know the relationships of sides and included angles from the rotation), \(\triangle ABC\cong\triangle EDC\)

Answer:

a. Because the rotation about \(C\) by angle \(ACE\) (the angle between \(CB\) and \(CD\) at \(C\)) moves \(CB\) to \(CD\) as rotation preserves the angle between segments at the center of rotation.
b. Because \(CB = CD\) (rotation preserves the distance from the center of rotation \(C\) to a point) and the direction of \(CB\) is rotated to the direction of \(CD\) (from part a).
c. Yes. A rotation is a rigid transformation that preserves side - lengths and angle - measures. So, by SSS (or SAS) congruence criterion, \(\triangle ABC\cong\triangle EDC\)