Question

what transformation transforms (a,b) to (a,-b)?
a reflection over the y - axis
a translation 1 unit up
a reflection over the x - axis
a translation 1 unit down
question 2
consider that △xyz is reflected across the y - axis and translated 2 units down.
choose all true statements about the transformation.
∠z≅∠z
△xyz is not congruent to △xyz.
xy≅xy
reflecting and translating the triangle change its shape and size.

Explanation:

Step1: Analyze point - transformation rules

In a coordinate - plane, for a point \((x,y)\), reflection over the y - axis changes the x - coordinate: \((x,y)\to(-x,y)\); translation 1 unit up changes the y - coordinate: \((x,y)\to(x,y + 1)\); reflection over the x - axis changes the y - coordinate: \((x,y)\to(x,-y)\); translation 1 unit down changes the y - coordinate: \((x,y)\to(x,y - 1)\). Since the transformation changes \((a,b)\) to \((a,-b)\), it is a reflection over the x - axis.

Step2: Analyze triangle - transformation properties

Reflections and translations are rigid motions. Rigid motions preserve side - lengths and angle - measures. For \(\triangle XYZ\) reflected across the y - axis and translated 2 units down:

• The measure of an angle is preserved under rigid motions, so \(\angle Z\cong\angle Z'\) (where \(Z'\) is the image of \(Z\)).
• Since rigid motions preserve congruence, \(\triangle XYZ\cong\triangle X'Y'Z'\).
• Side - lengths are preserved under rigid motions, so \(XY\cong X'Y'\).
• Reflecting and translating a triangle (rigid motions) do not change its shape and size.

Answer:

Question 1: a reflection over the x - axis
Question 2: \(\angle Z\cong\angle Z\), \(XY\cong XY\)