Question

triangle pqr is transformed according to the rule (x, y) → (x, -y) to create triangle pqr. image of grid with triangle which two statements are not true?
□ the line of reflection is the x-axis.
□ the side lengths and angle measures of triangle pqr are congruent to the corresponding side lengths and angle measures of triangle pqr.
□ the line of reflection is the y-axis.
□ the x-coordinate of triangle pqr is the same as the x-coordinate of triangle pqr and the y-coordinate triangle pqr is the opposite of the y-coordinate of triangle pqr
□ the side lengths of triangle pqr are congruent to the corresponding side lengths of triangle pqr, but the angle measures of triangle pqr are double the angle measures of triangle pqr.

Explanation:

To solve this, we analyze each statement based on the transformation rule \((x, y) \to (x, -y)\) (reflection over the \(x\)-axis):

Step 1: Analyze "The line of reflection is the \(x\)-axis."

A reflection over the \(x\)-axis has the rule \((x, y) \to (x, -y)\), which matches the given transformation. So this statement is true.

Step 2: Analyze "Side lengths/angles of \(P'Q'R'\) are congruent to \(PQR\)."

Reflections are rigid transformations (preserve side lengths and angle measures). So this statement is true.

Step 3: Analyze "The line of reflection is the \(y\)-axis."

A reflection over the \(y\)-axis has the rule \((x, y) \to (-x, y)\), which does not match \((x, -y)\). So this statement is not true.

Step 4: Analyze "The \(x\)-coordinate of \(P'Q'R'\) is the same as \(PQR\), and the \(y\)-coordinate of \(P'Q'R'\) is the opposite of \(PQR\)."

The transformation \((x, y) \to (x, -y)\) directly means \(x\)-coordinates stay the same, and \(y\)-coordinates are negated. So this statement is true.

Step 5: Analyze "Side lengths of \(P'Q'R'\) are congruent to \(PQR\), but angle measures of \(P'Q'R'\) are double those of \(PQR\)."

Reflections preserve both side lengths and angle measures (rigid transformation). Angle measures do not double. So this statement is not true.

Answer:

The two statements that are NOT true are:

• The line of reflection is the \(y\)-axis.
• The side lengths of triangle \(P'Q'R'\) are congruent to the corresponding side lengths of triangle \(PQR\), but the angle measures of triangle \(P'Q'R'\) are double the angle measures of triangle \(PQR\).