Reflecting shapes (article) | Reflections | Khan Academy (2024)

Learn how to find the image of a given reflection.

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  • Hannah

    6 years agoPosted 6 years ago. Direct link to Hannah's post “I understand how to algeb...”

    I understand how to algebraically perform reflections if the line of reflection is y = 0, x = 0, y = x, or y = -x. How can I algebraically perform a reflection for ANY line of reflection (e.g. how could I reflect (2, 9) across y = 7x + 2 algebraically)?

    (39 votes)

    • Ian Pulizzotto

      6 years agoPosted 6 years ago. Direct link to Ian Pulizzotto's post “Great question!Let A be...”

      Reflecting shapes (article) | Reflections | Khan Academy (4)

      Reflecting shapes (article) | Reflections | Khan Academy (5)

      Reflecting shapes (article) | Reflections | Khan Academy (6)

      Great question!

      Let A be the point to be reflected, let k be the line about which the point is reflected, let B represent the desired point (image), and let C represent the intersection of line k and line AB. Note that line AB must be perpendicular to line k, and C must be the midpoint of segment AB (from the definition of a reflection).
      So we can first find the equation of the line through point A that is perpendicular to line k. Then we can algebraically find point C, which is the intersection of these two lines. Then, using the fact that C is the midpoint of segment AB, we can finally determine point B.

      Example: suppose we want to reflect the point A(2,9) about the line k with equation y = 7x + 2. So we first find the equation of the line through (2,9) that is perpendicular to the line y = 7x + 2. Since the line y = 7x + 2 has slope 7, the desired line (that is, line AB) has slope -1/7 as well as passing through (2,9).

      So the desired line has an equation of the form y = (-1/7)x + b. Substituting the point (2,9) gives
      9 = (-1/7)(2) + b which gives b = 65/7. So the equation of this line is y = (-1/7)x + 65/7.

      Now we need to find the intersection of the lines y = 7x + 2 and y = (-1/7)x + 65/7 by solving this system of equations.
      Using the substitution method gives 7x + 2 = (-1/7)x + 65/7; (50/7)x = 51/7; x = 51/50.
      Then y = 7(51/50) + 2 = 457/50.

      So the intersection of the two lines is the point C(51/50, 457/50). Recall that A is the point (2,9).
      Since C is the midpoint of AB, we have
      B = C + (C - A) = (51/50 + 51/50 - 2, 457/50 + 457/50 - 9) = (1/25, 232/25).

      So the image (that is, point B) is the point (1/25, 232/25).

      (99 votes)

  • natalie.stringer22

    6 years agoPosted 6 years ago. Direct link to natalie.stringer22's post “I do not understand any o...”

    I do not understand any of this at all. Is there an easier way to learn/understand it?

    (43 votes)

  • louisaandgreta

    4 years agoPosted 4 years ago. Direct link to louisaandgreta's post “I could really use Sal ma...”

    I could really use Sal making a video about this, what’s written on this doc is really confusing.
    Sometimes they explain things that are pretty basic and other times more complicated things they’ll just assume that we know them even though we haven’t covered it/them yet.
    For instance I don’t understand what they mean when referring to the reflection line
    Y=1-x
    Y=x+2
    Y= x-5

    (28 votes)

    • yashaa

      2 years agoPosted 2 years ago. Direct link to yashaa's post “The reflection line is th...”

      The reflection line is the line that you are reflecting over. Y=mx+b is just the basic slope-intercept equation. If you don't understand slope -intercept, I recommend watching the videos Khan provides in the algebra courses. Since geometry tends to be taught after algebra in some cases, I think it's why they didn't explain it more in depth. Hope this helps!

      (9 votes)

  • McLachlin, Abigail

    a year agoPosted a year ago. Direct link to McLachlin, Abigail's post “I cried over this lesson ...”

    I cried over this lesson for over an hour, took a 2 day break,and then cried at the thought of it. I finally went back to it with a fresh head and realized I had over thought the whole things and it really wasn't that deep lol

    (26 votes)

    • 28jkim

      a year agoPosted a year ago. Direct link to 28jkim's post “wow that is bonkers”

      wow that is bonkers

      (5 votes)

  • rl0262

    6 years agoPosted 6 years ago. Direct link to rl0262's post “Is there a formula for th...”

    Is there a formula for the reflections?

    (22 votes)

    • 💲⚔💎💝Max Lennon💝💎⚔💲

      2 years agoPosted 2 years ago. Direct link to 💲⚔💎💝Max Lennon💝💎⚔💲's post “count the spaces between ...”

      count the spaces between the line you are reflecting over

      (4 votes)

  • andrewcwitt

    a year agoPosted a year ago. Direct link to andrewcwitt's post “Is there a more mathemati...”

    Is there a more mathematical way of calculating the reflection as opposed to manually counting on a graph? Perhaps using point slope (y=mx+b) or maybe by setting up a function? It seems like there should be a way to do this without requiring the graph.

    (8 votes)

    • connor N.

      a year agoPosted a year ago. Direct link to connor N.'s post “Yep, just plug the coordi...”

      Reflecting shapes (article) | Reflections | Khan Academy (23)

      Yep, just plug the coordinates for each point into the point-slope equation that is given to get the reflected points.

      Eg. In the last example in the article, you have the points M, O and N. To use M as an example, it's graphed at (-7,2). In the question, it tells you that it is reflected over a line of the equation y=-1-x

      To find the reflected coordinates of M using the point-slope equation given, plug in the following and solve:

      2=-1-x (Gives you the X coordinate)
      y=-1-(-7) (Gives you the Y coordinate)

      You can repeat this step for the other points (O and N) and then graph it.

      Hope that helps.

      (17 votes)

  • skylar.schrage

    4 years agoPosted 4 years ago. Direct link to skylar.schrage's post “why cant there be a video...”

    why cant there be a video on this i dont understand it but a video would help

    (14 votes)

    • ananyanair378

      4 years agoPosted 4 years ago. Direct link to ananyanair378's post “There is a part that says...”

      There is a part that says "I want to see Sal doing a similar question" which helped me since I was having trouble.

      (2 votes)

  • Anthony Senner

    2 years agoPosted 2 years ago. Direct link to Anthony Senner's post “Seriously, this math stum...”

    Seriously, this math stumps me

    (12 votes)

  • aoya joha

    3 years agoPosted 3 years ago. Direct link to aoya joha's post “isn't there an algebraic ...”

    isn't there an algebraic formula for this ?

    (4 votes)

    • Yagnesh Peddatimmareddy

      3 years agoPosted 3 years ago. Direct link to Yagnesh Peddatimmareddy's post “When you reflect a point ...”

      Reflecting shapes (article) | Reflections | Khan Academy (32)

      When you reflect a point across the line y = x, the x-coordinate and y-coordinate change places. If you reflect over the line y = -x, the x-coordinate and y-coordinate change places and are negated (the signs are changed). the line y = x is the point (y, x). the line y = -x is the point (-y, -x).

      (11 votes)

  • UmaFaLeung

    5 years agoPosted 5 years ago. Direct link to UmaFaLeung's post “This is a probably a stup...”

    This is a probably a stupid question, but i totally do not get why -- for example -- in this problem:

    Draw the image of triangle MNO under a reflection over: y = -1 -x

    I don't get what y = -1 -x means, is it a coordinate or is it comparing y to x, i just don't understand, same goes for the other problems:

    What is the image of (-12, 12) under a reflection over line y = x. I can usually solve the problem, but i feel like i still need to understand what is means.

    (5 votes)

    • Sarah

      5 years agoPosted 5 years ago. Direct link to Sarah's post “No questions are stupid! ...”

      No questions are stupid! y=-1-x and y=x are both lines. When you reflect a point, it is an equal distance away from the line as your original point. For instance, (-12,12) reflected over y=x would be (12,-12). I hope this clears things up!

      (9 votes)

Reflecting shapes (article) | Reflections | Khan Academy (2024)

FAQs

How do you reflect shapes easily? ›

To reflect an object, you need a mirror line. When a shape is reflected, its size does not change - the image just appears 'flipped'. Every point on the shape is the same distance away on the other side of the mirror line. Using squared paper can be very handy to help you reflect an object.

How to solve reflections? ›

To reflect over the x-axis, change the sign on the y coordinates of selected given points. Reflections over the y-axis require the x coordinated to be negated. Reflections over the origin require that both the x and y coordinates be negated, and to reflect over the line y=x, swap x and y.

How to find the line of reflection between two shapes? ›

A line of reflection is an imaginary line that flips one shape onto another. We find this line by finding the halfway points between matching points on the source and image triangles. All of the halfway points are on the line.

What does line of reflection mean in math? ›

A reflection is a mirror image of the shape. An image will reflect through a line, known as the line of reflection. A figure is said to reflect the other figure, and then every point in a figure is equidistant from each corresponding point in another figure.

What is an example of reflection of shapes? ›

The original shape or original image is called the pre-image and the reflected shape is called the image, reflected image, or mirror image. For example, Triangle P has been reflected across the line x = 4 x=4 x=4 to give Triangle. Q.

How to reflect over slope? ›

Step1: Determine the slope of a line perpendicular to the given line of reflection. Step 2: Draw a line perpendicular to the line of reflection through each point you want to reflect. Step 3: Determine the slope from each point to the point where the two perpendicular lines intersect.

What is the formula for reflection in geometry? ›

More generally, the image of any point (x, y) under reflection about the line y=b would be (x, 2b-y). Similarly, the image of any point (x, y) under reflection about the line x=a would be (2a-x, y). The concept of averaging in one coordinate and equality in the other coordinate leads to these formulas.

Which type of transformation changes the size of a shape or object? ›

The process of resizing or transforming an object is called dilation. It is a transformation that makes the objects smaller or larger with the help of the given scale factor.

How do you reflect a shape onto itself? ›

In order for the figure to map onto itself, the line of reflection must go through the center point. Two lines of reflection go through the sides of the figure. Two lines of reflection go through the vertices of the figure. Thus, there are four possible lines that go through the center and are lines of reflections.

How do you get a perfect reflection? ›

Some of the most beautiful reflections are found on the surface of water. Any bodies of water including tiny puddles are perfect for reflection photography. In addition to water, you can find great reflections on any glass surfaces, shiny cars, a wet tarmac, on ice or even your own sunglasses.

How do objects reflect? ›

Reflection of light (and other forms of electromagnetic radiation) occurs when the waves encounter a surface or other boundary that does not absorb the energy of the radiation and bounces the waves away from the surface.

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