Question

How do you find an axis of symmetry if only given points are $\left( {1, - 3} \right)$ and $\left( {9, - 3} \right)$ ?

Answer 1

Hint: The given points have the same y coordinate. This means if they are connected with a line, the line will be horizontal and will be a perpendicular bisector to the axis of symmetry. Hence, to find the axis of symmetry we need to find the midpoint of the line segment formed by these two points.

Formula used:
Midpoint Formula = $\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$

Complete step-by-step answer:
The axis of symmetry is a line that divides a figure or shape into two equal congruent halves. Hence, we can say the figure is symmetrical around the axis of symmetry
If a figure can be folded and if the two parts exactly match, then the folding line is known as the axis of symmetry.
Now, here the points given are $\left( {1, - 3} \right)$ and $\left( {9, - 3} \right)$ .
As we can see, the y coordinate of both points is the same.
Hence, they are said to be at the same horizontal level. And if we draw a line through these points, we get a horizontal line parallel to X-axis.
As both points are symmetrical to the axis of symmetry, we can say the axis of symmetry passes through the midpoint of the line segment connecting the points as shown in the figure.

Now, the formula to find the midpoint of a line segment is
$\Rightarrow$$\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$
Here, we can take the values of variables from the points as
$\Rightarrow$${x_1} = 1$ , ${x_2} = 9$ , ${y_1} = - 3$ , and ${y_2} = - 3$
Substituting the values,
$ \Rightarrow \left( {\dfrac{{1 + 9}}{2},\dfrac{{ - 3 - 3}}{2}} \right)$
$ \Rightarrow \left( {5, - 3} \right)$
Hence, we get the midpoint of the line as $\left( {5, - 3} \right)$ .
The axis of symmetry passes through $\left( {5, - 3} \right)$ and as we know, the equation of the axis of symmetry is obtained from the x coordinate
Hence, the equation of the axis of symmetry is

$ \Rightarrow x = 5$

Note: One must remember that if the x coordinate of both points is the same, the axis of symmetry is horizontal and its equation is obtained by considering the y coordinate of the midpoint. Similarly, if the y coordinate is the same, the equation is obtained from the x coordinate.