Question

the graph shows \\( \triangle abc \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\).
which sequence of transformations will carry \\( \triangle abc \\) onto \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\)?
a translation left 3 units and translation up 1 unit
b translation left 1 unit and translation down 3 units
c \\( 180 ^ { \circ } \\) clockwise rotation about the origin and a reflection about the \\( y \\)-axis
d \\( 180 ^ { \circ } \\) clockwise rotation about the origin and a reflection about the \\( y = x \\) line

Explanation:

Step1: Analyze translation

• For a translation, we check the change in \(x\) - coordinate (horizontal) and \(y\) - coordinate (vertical).
• Let's take a point, say \(A(-4,2)\) and \(A'(-5,-1)\). The change in \(x\) - coordinate: \(\Delta x=-5-(-4)=-1\) (left \(1\) unit). The change in \(y\) - coordinate: \(\Delta y=-1 - 2=-3\) (down \(3\) units).
• Check another point, \(B(-2,4)\) and \(B'(-3,1)\). \(\Delta x=-3-(-2)=-1\) (left \(1\) unit), \(\Delta y=1 - 4=-3\) (down \(3\) units).
• Check \(C(0,1)\) and \(C'(-1,-2)\). \(\Delta x=-1-0=-1\) (left \(1\) unit), \(\Delta y=-2 - 1=-3\) (down \(3\) units).

Step2: Analyze rotation and reflection

• For rotation and reflection:
• A \(180^{\circ}\) clock - wise rotation about the origin transforms a point \((x,y)\) to \((-x,-y)\). A reflection about the \(y\) - axis transforms \((x,y)\) to \((-x,y)\). A reflection about the \(y = x\) line transforms \((x,y)\) to \((y,x)\).
• If we consider rotation and reflection:
• Let's assume we first rotate \(\triangle ABC\) \(180^{\circ}\) clock - wise about the origin. \(A(-4,2)\) becomes \((4,-2)\), \(B(-2,4)\) becomes \((2,-4)\), \(C(0,1)\) becomes \((0,-1)\). Then reflecting about \(y\) - axis: \((4,-2)\) becomes \((-4,-2)

eq A'(-5,-1)\), \((2,-4)\) becomes \((-2,-4)
eq B'(-3,1)\), \((0,-1)\) becomes \((0,-1)
eq C'(-1,-2)\).

• If we first rotate \(180^{\circ}\) clock - wise about the origin and then reflect about \(y=x\) line: \(A(-4,2)\) after \(180^{\circ}\) rotation is \((4,-2)\), then after reflection about \(y = x\) is \((-2,4)

eq A'(-5,-1)\)

Answer:

B. translation left 1 unit and translation down 3 units