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Reflecting shapes

Learn how to find the image of a given reflection.
In this article we will find the images of different shapes under different reflections.

The line of reflection

A reflection is a transformation that acts like a mirror: It swaps all pairs of points that are on exactly opposite sides of the line of reflection.
The line of reflection can be defined by an equation or by two points it passes through.

Part 1: Reflecting points

Let's study an example of reflecting over a horizontal line

We are asked to find the image A, prime of A, left parenthesis, minus, 6, comma, 7, right parenthesis under a reflection over y, equals, 4.
A coordinate plane. The x- and y-axes both scale by one. A horizontal line passes through four on the y axis. A point A is at negative six, seven.

Solution

Step 1: Extend a perpendicular line segment from A to the reflection line and measure it.
Since the reflection line is perfectly horizontal, a line perpendicular to it would be perfectly vertical.
A coordinate plane. The x- and y-axes both scale by one. A horizontal line passes through four on the y axis. A point A is at negative six, seven. A ray extends from the point A three units to the dashed line.
Step 2: Extend the line segment in the same direction and by the same measure.
A coordinate plane. The x- and y-axes both scale by one. A horizontal line passes through four on the y axis. A point A is at negative six, seven. A ray extends from the point A three units to the dashed line. The ray passes the dashed line three more units.
Answer: A, prime is at left parenthesis, minus, 6, comma, 1, right parenthesis.
A coordinate plane. The x- and y-axes both scale by one. A horizontal line passes through four on the y axis. A point A is at negative six, seven. A ray extends from the point A three units to the dashed line. The ray passes the dashed line three more units pointing at the image of a point A prime at negative six, one.

Your turn!

Practice problem

Draw the image of B, left parenthesis, 7, comma, minus, 4, right parenthesis under a reflection over x, equals, 2.

Challenge problem

What is the image of left parenthesis, minus, 25, comma, minus, 33, right parenthesis under a reflection over the line y, equals, 0?
left parenthesis
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis

Let's study an example of reflecting over a diagonal line

We are asked to find the image C, prime of C, left parenthesis, minus, 2, comma, 9, right parenthesis under a reflection over y, equals, 1, minus, x.
A coordinate plane. The x- and y-axes both scale by one. A dashed line slants down from left to right passing through zero, one and one, zero. A point C is at negative two, nine.

Solution

Step 1: Extend a perpendicular line segment from C to the reflection line and measure it.
Since the reflection line passes exactly through the diagonals of the unit squares, a line perpendicular to it should pass through the other diagonal of the unit square. In other words, lines with slopes start text, 1, end text and start text, negative, 1, end text are always perpendicular.
For convenience, let's measure the distance in "diagonals":
A coordinate plane. The x- and y-axes both scale by one. A dashed line slants down from left to right passing through zero, one and one, zero. A point C is at negative two, nine. A ray extends from point C three diagonals to the dashed line.
Step 2: Extend the line segment in the same direction and by the same measure.
A coordinate plane. The x- and y-axes both scale by one. A dashed line slants down from left to right passing through zero, one and one, zero. A point C is at negative two, nine. A ray extends from point C three diagonals to the dashed line. The ray passes the dashed line and extends another three diagonals.
Answer: C, prime is at left parenthesis, minus, 8, comma, 3, right parenthesis.
A coordinate plane. The x- and y-axes both scale by one. A dashed line slants down from left to right passing through zero, one and one, zero. A point C is at negative two, nine. A ray extends from point C three diagonals to the dashed line. The ray passes the dashed line and extends another three diagonals pointing at the image point C prime at negative eight, three.

Your turn!

Practice problem

Draw the image of D, left parenthesis, 3, comma, minus, 5, right parenthesis under a reflection over y, equals, x, plus, 2.

Challenge problem

What is the image of left parenthesis, minus, 12, comma, 12, right parenthesis under a reflection over the line y, equals, x?
left parenthesis
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis

Part 2: Reflecting polygons

Let's study an example problem

Consider rectangle E, F, G, H drawn below. Let's draw its image E, prime, F, prime, G, prime, H, prime under a reflection over the line y, equals, x, minus, 5.
A coordinate plane. The x- and y-axes both scale by one. A dashed line slants up from left to right passing through zero, negative five and five, zero. A rectangle with the points E F G and H. Point E is at negative three, three, Point F is at five, three, Point G is at five, negative three. Point H is at negative three, negative three.

Solution

When we reflect a polygon, all we need is to perform the reflection on all of the vertices (this is similar to how we translate or rotate polygons).
Here are the original vertices and their images. Notice that E, F, and H were on an opposite side of the reflection line as G. The same is true about their images, but now they switched sides!
A coordinate plane. The x- and y-axes both scale by one. A dashed line slants up from left to right passing through zero, negative five and five, zero. Point E is at negative three, three, Point F is at five, three, Point G is at five, negative three. Point H is at negative three, negative three. A ray extends from point E through the dashed line to point E prime at eight, negative eight. A ray extends from point F through the dashed line to point F prime at eight, zero. A ray extends from point G through the dashed line to point G prime at two, zero. A ray extends from point H through the dashed line to point H prime at two, negative eight.
Now we simply connect the vertices.
A coordinate plane. The x- and y-axes both scale by one. A dashed line slants up from left to right passing through zero, negative five and five, zero. Point E is at negative three, three, Point F is at five, three, Point G is at five, negative three. Point H is at negative three, negative three. A ray extends from point E through the dashed line to point E prime at eight, negative eight. A ray extends from point F through the dashed line to point F prime at eight, zero. A ray extends from point G through the dashed line to point G prime at two, zero. A ray extends from point H through the dashed line to point H prime at two, negative eight. E prime, F prime, G prime, and H prime form the image of a rectangle.

Your turn!

Problem 1

Draw the images of the line segments start overline, I, J, end overline and start overline, K, L, end overline under a reflection over y, equals, minus, 3.

Problem 2

Draw the image of triangle, M, N, O under a reflection over y, equals, minus, 1, minus, x.

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