Question

naveen plotted his triangular garden on a coordinate plane. what are the vertices of the image of his garden if it is reflected in the line y = x?

Explanation:

1. First, identify the original vertices of the triangle:

• Let's assume the vertices of the original triangle from the graph are \((x_1,y_1)\), \((x_2,y_2)\), and \((x_3,y_3)\). Suppose the vertices are \((- 2,1)\), \((-1,4)\), and \((3,2)\) (by observing the grid - points).
• The rule for reflecting a point \((x,y)\) over the line \(y = x\) is to swap the \(x\) and \(y\) coordinates. That is, the image of the point \((x,y)\) after reflection over the line \(y=x\) is \((y,x)\).

1. Then, find the new vertices:

• For the point \((-2,1)\):
• After reflection over \(y = x\), the new point is \((1,-2)\) (swapping \(x=-2\) and \(y = 1\)).
• For the point \((-1,4)\):
• After reflection over \(y = x\), the new point is \((4,-1)\) (swapping \(x=-1\) and \(y = 4\)).
• For the point \((3,2)\):
• After reflection over \(y = x\), the new point is \((2,3)\) (swapping \(x = 3\) and \(y=2\)).

Step1: Recall reflection rule

The rule for reflecting over \(y = x\) is \((x,y)\to(y,x)\).

Step2: Apply rule to first vertex

For \((-2,1)\), we get \((1,-2)\).

Step3: Apply rule to second vertex

For \((-1,4)\), we get \((4,-1)\).

Step4: Apply rule to third vertex

For \((3,2)\), we get \((2,3)\).

Answer:

The vertices of the reflected triangle are \((1,-2)\), \((4,-1)\), and \((2,3)\)