The equation for the line of reflection which maps the trapezoid onto itself is [tex]\fbox{\begin\\\ \math y=2\\\end{minispace}}[/tex].
Further explanation:
From the given figure it is observed that there is a trapezoid placed vertically and the corner points of the trapezoid are [tex](-2,7),(-2,-3),(2,0)[/tex] and [tex](2,4)[/tex].
Label the point [tex](-2,7)[/tex] as A, [tex](-2,-3)[/tex] as B, [tex](2,0)[/tex] as C and [tex](2,4)[/tex] 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 [tex]10\ \text{units}[/tex], length of side CD is [tex]4\ \text{units}[/tex].
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 [tex]10\ \text{units}[/tex] and the length of the side CD is [tex]4\ \text{units}[/tex].
The coordinate of point A is [tex](-2,7)[/tex] and the coordinate for point B is [tex](-2,-3)[/tex].
As per the mid-point theorem the mid-point of a line joining the point [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] is calculated as follows:
[tex]\fbox{\begin\\\ \math (x,y)=\left(\frac{(x_{1}+x_{2})}{2},\frac{(y_{1}+y_{2})}{2}\right)\\\end{minispace}}[/tex]
The coordinate of midpoint of AB is calculated as follows:
[tex]\begin{aligned}(x,y)&=\left(\frac{-2-2}{2},\frac{7-3}{1}\right)\\&=(-2,2)\end{aligned}[/tex]
Therefore, the coordinate of midpoint of AB is [tex](-2,2)[/tex].
The coordinate of point C is [tex](2,0)[/tex] and the coordinate for point D is [tex](2,4)[/tex].
The coordinate of midpoint of CD is calculated as follows:
[tex]\begin{aligned}(x,y)&=\left(\frac{2+2}{2},\frac{0+4}{2}\right)\\&=(2,2)\end{aligned}[/tex]
Therefore, the coordinate of midpoint of CD is [tex](2,2)[/tex].
From the above calculation and the given graph it is concluded that the line [tex]y=2[/tex] 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 [tex]\triangle \text{ADE}[/tex].
By using the Pythagoras theorem the length of AD is calculated as follows:
[tex]\begin{aligned}\text{AD}&=\sqrt{4^{2}+3^{2}}\\&=\sqrt{16+9}\\&=\sqrt{25}\\&=5\end{aligned}[/tex]
Similarly, for [tex]\triangle \text{BCF}[/tex] the length of BC is calculated as follows:
[tex]\begin{aligned}\text{BC}&=\sqrt{4^{2}+3^{2}}\\&=\sqrt{16+9}\\&=\sqrt{25}\\&=5\end{aligned}[/tex]
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 [tex]y=2[/tex] 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 [tex]y=2[/tex] is the line of reflection for the trapezoid ABCD.
From figure 4 it is observed that the line [tex]y=2[/tex] is the line of symmetry for the trapezoid ABCD.
Therefore, the equation of line of reflection for the trapezoid is [tex]\fbox{\begin\\\ \math y=2\\\end{minispace}}[/tex].
Learn more:
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2. A problem to determine the slope intercept form of a line https://brainly.com/question/1473992
3. Inverse function https://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.
