Question

name each image.
a. ( r ( 180 ^ { circ } , ( 0,1 ) ) ( \triangle a b c ) )
b. ( ( t ( 0,8 ) circ r _ { y - \text { axis } } ) ( \triangle t u v ) )
c. ( ( t ( - 2,2 ) circ r _ { x - \text { axis } } ) ( \triangle d f e ) )
d. ( r _ { y - \text { axis } } ( \triangle t u v ) )
a. ( r ( 180 ^ { circ } , ( 0,1 ) ) ( \triangle a b c ) = \triangle square )

Explanation:

Step1: Understand rotation concept

Rotation of \(180^{\circ}\) about a point \((h,k)\) has the transformation rule \((x,y)\to(2h - x,2k - y)\). When rotating about the origin \((0,0)\), the rule is \((x,y)\to(-x,-y)\). Here, rotating about \((0,1)\), so the rule is \((x,y)\to(-x,2 - y)\).

Step2: Analyze the triangles

For \(\triangle ABC\) and its image after \(R(180^{\circ},(0,1))\), we check the vertices. Let's assume a vertex \(A(x_1,y_1)\) of \(\triangle ABC\). After rotation about \((0,1)\), its image \(A'( - x_1,2 - y_1)\). By observing the grid (counting the units of translation for vertices), we can match \(\triangle ABC\) with \(\triangle XYZ\) (assuming \(X,Y,Z\) are the vertices of the image triangle).

Answer:

\(\triangle XYZ\)