Question

Find the image of point $\left( 1,7 \right)$ in x-axis and y-axis.

Answer 1

Hint: For solving these problems, we need to have a clear understanding of the graphical method of representing points and the properties of reflection of points on the respective axes. By employing these rules of reflection, we can easily find out the point when it gets reflected on the x-axis or the y-axis.

Complete step-by-step solution:
Two numbers that represent a point on a graph, are also called the coordinates. Abscissa - The horizontal line, or x-axis, of a graph. Ordinate - The vertical line, or y-axis, of a graph. Origin - The origin is the point where the X and Y axis intersect on a graph. This is the point $\left( 0,0 \right)$ in a two-dimensional graph. The x-coordinate of a point is the value that tells you how far from the origin the point is on the horizontal, or x-axis. The y-coordinate of a point is the value that tells you how far from the origin the point is on the vertical, or y-axis.
To find the coordinates, the x-axis represents the plain mirror. M is the point in the rectangular axes in the first quadrant whose coordinates are $\left( h,k \right)$ . When point M is reflected in the x-axis, the image M’ is formed in the fourth quadrant whose coordinates are $\left( h,-k \right)$ . Thus, we conclude that when a point is reflected in the x-axis, then the x-co-ordinate remains the same, but the y-coordinate becomes negative. Thus, the image of point $M\left( h,k \right)$ is $M'\left( h,-k \right)$ . Thus, the reflection of the point $\left( 1,7 \right)$ , lying in the first quadrant, in the x-axis will be $\left( 1,-7 \right)$ .
To find the coordinates, the y-axis represents the plain mirror. N is the point in the rectangular axes in the first quadrant whose coordinates are $\left( h,k \right)$ . When point N is reflected in the y-axis, the image N’ is formed in the second quadrant whose coordinates are $\left( -h,k \right)$ . Thus, we conclude that when a point is reflected in the y-axis, then the y-coordinate remains the same, but the x-co-ordinate becomes negative. Thus, the image of point $N\left( h,k \right)$ is $N'\left( -h,k \right)$ . Thus, the reflection of the point $\left( 1,7 \right)$ , lying in the first quadrant, in the y-axis will be $\left( -1,7 \right)$ .
Hence, the image of the point $\left( 1,7 \right)$ after being reflected in the x and y axes respectively are $\left( -1,-7 \right)$ and $\left( -1,7 \right)$ .

Note: These types of problems are pretty easy to solve, but one needs to be careful otherwise small misjudgements can lead to a totally different answer. We need to carefully employ the properties of the reflection of the points in x and y axes to arrive at the correct answer and get the correct image.