Question

triangle rst has vertices r(2, 0), s(4, 0), and t(1, −3). the image of triangle rst after a rotation has vertices r(0, −2), s(0, −4), and t(−3, −1). which rule describes the transformation?
○ ( r_{0, 90^circ} )
○ ( r_{0, 180^circ} )
○ ( r_{0, 270^circ} )
○ ( r_{0, 360^circ} )

Explanation:

Step1: Recall rotation rules

The rotation rules about the origin \( O(0,0) \) are:

• \( 90^\circ \) rotation: \( (x,y)\to(-y,x) \)
• \( 180^\circ \) rotation: \( (x,y)\to(-x,-y) \)
• \( 270^\circ \) rotation: \( (x,y)\to(y,-x) \)
• \( 360^\circ \) rotation: \( (x,y)\to(x,y) \) (identity transformation)

Step2: Test \( 270^\circ \) rotation on point \( R(2,0) \)

Using the \( 270^\circ \) rotation rule \( (x,y)\to(y,-x) \):
For \( R(2,0) \), substitute \( x = 2 \), \( y = 0 \):
\( (0,- 2) \), which matches \( R'(0,-2) \).

Step3: Test \( 270^\circ \) rotation on point \( S(4,0) \)

For \( S(4,0) \), substitute \( x = 4 \), \( y = 0 \) into \( (y,-x) \):
\( (0,-4) \), which matches \( S'(0,-4) \).

Step4: Test \( 270^\circ \) rotation on point \( T(1,-3) \)

For \( T(1,-3) \), substitute \( x = 1 \), \( y=-3 \) into \( (y,-x) \):
\( (-3,-1) \), which matches \( T'(-3,-1) \).

Step5: Verify other rotations (optional)

• \( 90^\circ \) rotation on \( R(2,0) \): \( (0,2)

eq(0,-2) \)

• \( 180^\circ \) rotation on \( R(2,0) \): \( (-2,0)

eq(0,-2) \)

• \( 360^\circ \) rotation on \( R(2,0) \): \( (2,0)

eq(0,-2) \)

Answer:

\( R_{0,270^\circ} \)