Question

is pqr similar to uvw? justify your answer.
yes, because a reflection over the x - axis followed by a translation left 10 units and up 2 units maps pqr onto uvw.
yes, because a rotation 90° counterclockwise around the origin followed by a translation down 7 units maps pqr onto uvw.
no, because ∠p and ∠u do not have the same measure.

Explanation:

Step1: Analyze the first option

A reflection over the \(x -\)axis changes the \(y -\)coordinate sign. A translation left \(10\) units subtracts \(10\) from the \(x -\)coordinate and up \(2\) units adds \(2\) to the \(y -\)coordinate. If we assume coordinates of \(PQR\) (say \(Q(2,7)\), \(R(4,3)\) and another point) and apply the transformation: for \(Q(2,7)\), after reflection over \(x -\)axis \((2,- 7)\), then translation \((2 - 10,-7 + 2)=(-8,-5)\) which is not a point of \(UVW\).

Step2: Analyze the second option

A \(90^{\circ}\) counter - clockwise rotation around the origin transforms a point \((x,y)\) to \((-y,x)\). Then a translation down \(7\) units subtracts \(7\) from the \(y -\)coordinate. If we assume coordinates of \(PQR\) (say \(Q(2,7)\), \(R(4,3)\) and another point). For \(Q(2,7)\), after \(90^{\circ}\) counter - clockwise rotation \((-7,2)\), then translation \((-7,2-7)=(-7,-5)\) which is not a point of \(UVW\).

Step3: Analyze the third option

Similar triangles have congruent corresponding angles. If \(\angle P\) and \(\angle U\) do not have the same measure, then the triangles are not similar.

Answer:

No, because \(\angle P\) and \(\angle U\) do not have the same measure.