Question

graph the image of \\( \triangle tuv \\) after the following sequence of transformations:
rotation \\( 180 ^ { \circ } \\) counterclockwise around the origin
translation 14 units right and 9 units down

Explanation:

Step1: Find coordinates of original points

Assume \( T(6,3)\), \( U(8, - 3)\), \( V(12,3)\)

Step2: Apply rotation \(180^{\circ}\) counter - clockwise

The rule for a \(180^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-x,-y)\)

• For \(T(6,3)\): \((6,3)\to T_1(-6,-3)\)
• For \(U(8,-3)\): \((8,-3)\to U_1(-8,3)\)
• For \(V(12,3)\): \((12,3)\to V_1(-12,-3)\)

Step3: Apply translation \((x,y)\to(x + 14,y-9)\)

• For \(T_1(-6,-3)\): \((-6+14,-3 - 9)=(8,-12)\)
• For \(U_1(-8,3)\): \((-8 + 14,3-9)=(6,-6)\)
• For \(V_1(-12,-3)\): \((-12+14,-3-9)=(2,-12)\)

Answer:

The new coordinates of the vertices of \(\triangle TUV\) after the sequence of transformations are \(T'(8,-12)\), \(U'(6,-6)\), \(V'(2,-12)\)