Given a quadrilateral having coordinates A(6,8), B(12,4), C(6,0) and D(4,0), first graph the quadrilateral on graph paper and connect the vertices. Shade in the quadrilateral and then reflect it across the x-axis. Do NOT shade in the resulting image. Now reflect this new image across the line y=-14 and shade in the final image. Explain what the same is and what is different about the two shaded quadrilaterals(the original one and the final one). What is the relationship between the two lines of reflection? Do you think that makes a difference? Why or why not? Please help me with this and thank you to helps me.
Can someone help me with this? I don't understand how to do the y=-14 part my teacher didn't explain it to us today.
Obviously I cannot help you with the graphing, but
I assume you can do the reflection about the x-axis
(any point on the x-axis would stay, all points keep their x's, and their y's become opposite.
e.g. A(6,8) ---> A1(6,-8)
repeat for the other points and graph the new figure, then shade as instructed.
y = -14 is simply a horizontal line 14 units below the x-axis (two points on it would be (5,-14) and (-5,-14) , join them)
Now you are reflecting all points around that line
the x's would stay the same, but for the y, calculate how far above -14 it is, then move down just as far below the -14
e.g. B(12,4) is 18 units above the line y = -14
(how did I get 18 ??)
so it moves 18 units below that line, as a result:
B(12,4) ---> B2(12,-32)
you can check your new points are correct by taking the average of your y values of the old point and the new point, you should get -14
(-32 + 4)/2 = -14
give it a try, I suggest you print out my instructions.
I don't get it
To graph the quadrilateral, plot the four given coordinates A(6,8), B(12,4), C(6,0), and D(4,0) on graph paper. Connect the vertices in the order they are given.
Next, to reflect the quadrilateral across the x-axis, imagine flipping the figure over the x-axis. The coordinates of the reflected quadrilateral will have the same x-coordinates but opposite y-coordinates. For example, A(6,8) would reflect to A'(6,-8), B(12,4) would reflect to B'(12,-4), C(6,0) would reflect to C'(6,0) (since the y-coordinate is already on the x-axis), and D(4,0) would reflect to D'(4,0) as well.
To reflect the new image across the line y = -14, imagine flipping the figure over this line. The coordinates of the reflected quadrilateral will be the same as the x-axis reflection but with a change in y-coordinate. The line y = -14 is a horizontal line 14 units below the x-axis. So, to reflect a point across this line, you move the same distance below (-14 - y-coordinate). For example, A'(6,-8) would reflect to A''(6,-22), B'(12,-4) would reflect to B''(12,-18), C'(6,0) would reflect to C''(6,-14), and D'(4,0) would reflect to D''(4,-14).
Note that when reflecting across the x-axis or a horizontal line, the x-coordinate remains the same, but the sign of the y-coordinate is changed. When reflecting across the line y = -14, the y-coordinate will change but the x-coordinate remains the same.
Regarding the difference between the two shaded quadrilaterals, the main difference is the orientation. In the original quadrilateral, the top side is horizontal, while in the final quadrilateral, the bottom side is horizontal. The original quadrilateral is above the x-axis, while the final quadrilateral is below the x-axis. Additionally, the original quadrilateral is not shaded, but the final quadrilateral is shaded in.
The relationship between the two lines of reflection is that they are both horizontal reflections but occur on different lines. The x-axis reflection flips the figure over the x-axis, while the reflection across the line y = -14 flips the figure over a different horizontal line parallel to the x-axis. The order in which these reflections are performed can affect the final outcome, as reversing the order would result in a different figure.
Sure, I'd be happy to help you with this problem. Let's break it down step by step.
Step 1: Graph the original quadrilateral
To graph the original quadrilateral ABCD, plot the four given points A(6, 8), B(12, 4), C(6, 0), and D(4, 0) on a graph paper. Connect the vertices with straight lines to form the quadrilateral.
Step 2: Shade in the original quadrilateral
Once you have connected the vertices, shade in the interior of the quadrilateral with a pencil or pen. This represents the original shape.
Step 3: Reflect the original quadrilateral across the x-axis
To reflect the original quadrilateral across the x-axis, you need to "flip" the entire shape vertically. This means that each point's y-coordinate will be negated while the x-coordinate remains the same.
For example, point A(6, 8) becomes A'(6, -8) after reflecting it across the x-axis. Similarly, point B(12, 4) becomes B'(12, -4), and so on. Connect the reflected points A', B', C', and D' to form the reflected quadrilateral.
Note that the reflected quadrilateral will be below the x-axis (since the y-coordinates are negated), but we do not shade it in yet.
Step 4: Reflect the reflected quadrilateral across the line y = -14
To reflect the reflected quadrilateral across the line y = -14, you need to "flip" the entire shape horizontally. This means that each point's x-coordinate will change while the y-coordinate remains the same.
To find the reflected coordinates of each point, you can imagine the line y = -14 as a mirror. The distance between each point and the line y = -14 will be the same for the original point and its reflected image.
For example, for point A', imagine a line perpendicular to y = -14 passing through A'. This line intersects y = -14 at a certain point, and let's call it A''. The coordinates of A'' will be the same as A' except with the y-coordinate of -14.
Once you have found the reflected coordinates of all the points (A'', B'', C'', D''), connect them to form the final shape.
Step 5: Shade in the final quadrilateral
Now that you have the final quadrilateral after reflecting it across the line y = -14, shade in its interior to complete the task.
Now, let's answer your questions:
What is the same and what is different about the two shaded quadrilaterals?
The same: Both shaded quadrilaterals have the same shape. They have the same side lengths and vertex angles.
The difference: The two shaded quadrilaterals are mirror images of each other. The orientation of the quadrilaterals is reversed.
What is the relationship between the two lines of reflection?
The first reflection is across the x-axis, and the second reflection is across the line y = -14. These two lines are perpendicular to each other.
Does the relationship between the two lines of reflection make a difference?
Yes, it does make a difference. If the two lines of reflection were parallel or at an angle other than 90 degrees, the resulting image after the second reflection would be different. The perpendicular relationship ensures that the final image is a perfect mirror reflection of the original shape.
I hope this clarifies the process and answers your questions. If you have any further doubts, feel free to ask!