Question

given ( m(\triangle pqr)=\triangle pqr ), do ( \triangle pqr ) and ( \triangle pqr ) have equal perimeters? explain.
a is a rigid motion, so side lengths preserved. this means ( \triangle pqr ) and ( \triangle pqr ) equal perimeters.

Explanation:

Step1: Understand rigid motion

Rigid motion (like translation, rotation, reflection) keeps side - lengths unchanged.

Step2: Recall perimeter formula

Perimeter of a triangle \(P = a + b + c\) (where \(a,b,c\) are side - lengths).

Step3: Compare perimeters

Since \(m(\triangle PQR)=m(\triangle P'Q'R')\), side - lengths \(PQ = P'Q'\), \(QR = Q'R'\), \(PR = P'R'\). Then perimeter of \(\triangle PQR=PQ + QR+PR\) and perimeter of \(\triangle P'Q'R'=P'Q' + Q'R'+P'R'\). Substituting \(PQ = P'Q'\), \(QR = Q'R'\), \(PR = P'R'\) into the perimeter formulas, we get that the perimeters are equal.

Answer:

Yes, \(\triangle P'Q'R'\) and \(\triangle PQR\) have equal perimeters. Because rigid motion preserves side - lengths. If \(PQ = P'Q'\), \(QR = Q'R'\), \(PR = P'R'\), and perimeter of \(\triangle PQR=PQ + QR + PR\), perimeter of \(\triangle P'Q'R'=P'Q' + Q'R'+P'R'\), by substitution, their perimeters are equal.