Question

translate △def to the right 5 units and up 7 units. what are the coordinates of △def?

solution d(0,5) e(5,5) f(1,1)

figures a and b are congruent. which sequences of transformations can you perform on figure a to show this? choose all the correct answers.

reflect figure a across the y - axis.
rotate figure a 180° counterclockwise.
reflect figure a across the y - axis. then rotate figure a 180° clockwise.
reflect figure a across the x - axis. then rotate figure a 180° clockwise.
rotate figure a 180° clockwise. then reflect it across the x - axis.

Explanation:

Step1: Recall translation rule

For a translation to the right by \(h\) units and up by \(k\) units, the rule for a point \((x,y)\) is \((x + h,y + k)\). Here \(h = 5\) and \(k=7\). But since the original coordinates of \(\triangle DEF\) are not given clearly in the visible - part of the problem, we assume we are just verifying the given answer based on the rule.

Step2: Analyze the congruence problem

To show two figures \(A\) and \(B\) are congruent, we use transformation rules.

• Reflecting figure \(A\) across the \(y\) - axis changes the sign of the \(x\) - coordinate (\((x,y)\to(-x,y)\)).
• Rotating a figure \(180^{\circ}\) counter - clockwise or clockwise about the origin changes the signs of both coordinates (\((x,y)\to(-x,-y)\)).
• Reflecting across the \(x\) - axis changes the sign of the \(y\) - coordinate (\((x,y)\to(x, - y)\)).
• For figure \(A\) and \(B\), reflecting figure \(A\) across the \(y\) - axis will map it onto figure \(B\) as they are mirror images with respect to the \(y\) - axis.

Answer:

Reflect figure \(A\) across the \(y\) - axis.