Question

What is the equation for the line of reflection that maps the trapezoid onto itself?

Answer 1

The equation for the line of reflection which maps the trapezoid onto itself is $\fbox{\begin\\\ \math y=2\\\end{minispace}}$.

Further explanation:

From the given figure it is observed that there is a trapezoid placed vertically and the corner points of the trapezoid are $(-2,7),(-2,-3),(2,0)$ and $(2,4)$.

Label the point $(-2,7)$ as A, $(-2,-3)$ as B, $(2,0)$ as C and $(2,4)$ as D.

Figure 1 (attached in the end) shows the trapezoid ABCD.

From the given graph it is observed that the length of the side AB is $10\ \text{units}$, length of side CD is $4\ \text{units}$.

The line of reflection is a line which reflects the image of an object onto the other side in such a way that the reflected image is same as the original object.

For a regular polygon the line of reflection is the line of symmetry.

This implies that the line of symmetry for the given trapezium is its line of reflection.

A line of symmetry of a figure is a line which divides a figure into exactly two halves such that they can even overlap with each other.

For the given trapezium if we consider the line of symmetry as a vertical line then it is observed that any vertical line which cuts the trapezoid in two parts never gives two symmetrical halves.

From figure 2 (attached in the end) it is observed that any vertical does cut the trapezoid into two symmetrical halves.

The length of the side AB is $10\ \text{units}$ and the length of the side CD is $4\ \text{units}$.

The coordinate of point A is $(-2,7)$ and the coordinate for point B is $(-2,-3)$.

As per the mid-point theorem the mid-point of a line joining the point $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is calculated as follows:

$\fbox{\begin\\\ \math (x,y)=\left(\frac{(x_{1}+x_{2})}{2},\frac{(y_{1}+y_{2})}{2}\right)\\\end{minispace}}$

The coordinate of midpoint of AB is calculated as follows:

$\begin{aligned}(x,y)&=\left(\frac{-2-2}{2},\frac{7-3}{1}\right)\\&=(-2,2)\end{aligned}$

Therefore, the coordinate of midpoint of AB is $(-2,2)$.  

The coordinate of point C is $(2,0)$ and the coordinate for point D is $(2,4)$.

The coordinate of midpoint of CD is calculated as follows:

$\begin{aligned}(x,y)&=\left(\frac{2+2}{2},\frac{0+4}{2}\right)\\&=(2,2)\end{aligned}$

Therefore, the coordinate of midpoint of CD is $(2,2)$.

From the above calculation and the given graph it is concluded that the line $y=2$ divides the sides AB and CD into two halves.

From figure 3 (attached in the end) it is observed a right angle triangle is formed as $\triangle \text{ADE}$.

By using the Pythagoras theorem the length of AD is calculated as follows:

$\begin{aligned}\text{AD}&=\sqrt{4^{2}+3^{2}}\\&=\sqrt{16+9}\\&=\sqrt{25}\\&=5\end{aligned}$

Similarly, for $\triangle \text{BCF}$ the length of BC is calculated as follows:

$\begin{aligned}\text{BC}&=\sqrt{4^{2}+3^{2}}\\&=\sqrt{16+9}\\&=\sqrt{25}\\&=5\end{aligned}$

This implies that the sides AD and BC are equal in length.

Since, the length of the side AD and BC are equal so the line $y=2$ will divide the trapezoid into two symmetrical halves in such a way that the two halves completely overlap with each other.

As stated above line of reflection for a regular polygon is its line of symmetry.

This implies that the line $y=2$ is the line of reflection for the trapezoid ABCD.

From figure 4 it is observed that the line $y=2$ is the line of symmetry for the trapezoid ABCD.

Therefore, the equation of line of reflection for the trapezoid is $\fbox{\begin\\\ \math y=2\\\end{minispace}}$.

Learn more:  

1. A problem to complete the square of quadratic function brainly.com/question/12992613  

2. A problem to determine the slope intercept form of a line brainly.com/question/1473992

3. Inverse function brainly.com/question/1632445  

Answer details  

Grade: Middle school  

Subject: Mathematics  

Chapter: Coordinate geometry

Keywords: Geometry, coordinate geometry, reflection, symmetry, polygon, line of reflection, line of symmetry, mid-point theorem, trapezoid, map trapezoid onto, equation, y=2, regular polygon, reflection of trapezoid.

Answer 2

Answer:

x = 2

Explanation:

A line of reflection that maps a polygon onto itself will be a line of reflectional symmetry in the figure.

A line of reflectional symmetry is a line through which a figure can be folded in half.  For this trapezoid, a line of reflectional symmetry must be a perpendicular bisector of both bases.

This means in this figure, the line must be horizontal and must run at x = 2.