Question

one vertex of a triangle is located at (0, 5) on a coordinate grid after a transformation, the vertex is located at (5, 0)
which transformations could have taken place? select two options
□ r₀, 90°
□ r₀, 180°
□ r₀, 270°
□ r₀, -90°
□ r₀, -180°

Explanation:

Step1: Recall the rotation rules

• For a point \((x,y)\) rotated \(90^{\circ}\) counter - clockwise about the origin \((0,0)\): \((x,y)\to(-y,x)\)
• For a point \((x,y)\) rotated \(180^{\circ}\) counter - clockwise about the origin \((0,0)\): \((x,y)\to(-x,-y)\)
• For a point \((x,y)\) rotated \(270^{\circ}\) counter - clockwise about the origin \((0,0)\): \((x,y)\to(y,-x)\)
• For a point \((x,y)\) rotated \(90^{\circ}\) clockwise about the origin \((0,0)\) (equivalent to \(R_{0,-90^{\circ}}\)): \((x,y)\to(y,-x)\)
• For a point \((x,y)\) rotated \(180^{\circ}\) clockwise about the origin \((0,0)\) (equivalent to \(R_{0,-180^{\circ}}\)): \((x,y)\to(-x,-y)\)

Step2: Apply the rules to the given point

• Given the original point \((0,5)\)
• For \(R_{0,270^{\circ}}\):
• Using the rule \((x,y)\to(y,-x)\), when \(x = 0\) and \(y = 5\), we get \((5,0)\)
• For \(R_{0,-90^{\circ}}\) (clockwise \(90^{\circ}\) rotation):
• Using the rule \((x,y)\to(y,-x)\), when \(x = 0\) and \(y = 5\), we get \((5,0)\)
• For \(R_{0,90^{\circ}}\):
• Using the rule \((x,y)\to(-y,x)\), when \(x = 0\) and \(y = 5\), we get \((-5,0)\)
• For \(R_{0,180^{\circ}}\):
• Using the rule \((x,y)\to(-x,-y)\), when \(x = 0\) and \(y = 5\), we get \((0,-5)\)
• For \(R_{0,-180^{\circ}}\):
• Using the rule \((x,y)\to(-x,-y)\), when \(x = 0\) and \(y = 5\), we get \((0,-5)\)

So the transformations that could have taken place are \(R_{0,270^{\circ}}\) and \(R_{0,-90^{\circ}}\)

Answer:

C. \(R_{0,270^{\circ}}\), D. \(R_{0, - 90^{\circ}}\)