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Reflections

In the axes, y = x, x = k and y = k · Geometry · 21 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Draw the shape
  2. Draw the mirror line
  3. Reflect each point
  4. The new point
  5. Check

The steps for Reflect (3, 2) in the x-axis. Every solution names its steps, so you always know what comes next.

Worked examples

Reflect (3, 2) in the x-axis Easy

The question

Question: reflect the point P(3, 2) in the x-axis.

P = (3, 2)
= in the x-axis

1 Draw the shape

Put the point on the grid.

2 Draw the mirror line

The mirror line is the x-axis.

Reflecting flips the shape over the line, like a mirror.

Each point lands just as far away, on the other side.

reflection = flip

3 Reflect each point

From P, count straight to the mirror line: 2.

Then count 2 more on the other side.

We call the new point P prime.

The little mark means: the new P.

P is 2 above the x-axis.

Its image is 2 below it.

Its x stays 3.

So P' is (3, −2).

P(3, 2) → P'(3, −2)

Careful!

The x-axis is flat.

The flip goes up or down, so y changes.

x stays.

Careful: not P'(−3, 2) ✗

4 The new point

So the image is P'(3, −2).

P' = (3, −2)

5 Check

Let's check with the rule for the x-axis: x stays the same, y changes sign.

x-axis: (x, y) → (x, −y)

It gives the same points.

Right!

P(3, 2) → P'(3, −2)

Let's look back.

First: draw the shape.

Then: draw the mirror line.

Then: reflect each point.

Then: the new point.

Last: check.

Answer
P' = (3, −2)

▶ Watch it solved

Reflect (1, 4) in the line y = x Medium

The question

Question: reflect the point P(1, 4) in the line y = x.

P = (1, 4)
= in the line y = x

1 Draw the shape

Put the point on the grid.

2 Draw the mirror line

The mirror line is the line y = x.

It goes through (0, 0), (1, 1), (2, 2) and so on.

Reflecting flips the shape over the line, like a mirror.

Each point lands just as far away, on the other side.

reflection = flip

3 Reflect each point

From P, go straight to the mirror line, at a right angle to it.

Then go the same again, past the line.

We call the new point P prime.

The little mark means: the new P.

P is (1, 4).

Across the mirror line, P' is (4, 1).

P(1, 4) → P'(4, 1)

Careful!

Over the line y = x, the two numbers swap places.

Their signs do not change.

Careful: not P'(−4, −1) ✗

4 The new point

So the image is P'(4, 1).

P' = (4, 1)

5 Check

Let's check with the rule for the line y = x: x and y swap places.

y = x: (x, y) → (y, x)

It gives the same points.

Right!

P(1, 4) → P'(4, 1)

Let's look back.

First: draw the shape.

Then: draw the mirror line.

Then: reflect each point.

Then: the new point.

Last: check.

Answer
P' = (4, 1)

▶ Watch it solved

Reflect (2, 3) in the line x = 4 Harder

The question

Question: reflect the point P(2, 3) in the line x = 4.

P = (2, 3)
= in the line x = 4

1 Draw the shape

Put the point on the grid.

2 Draw the mirror line

The mirror line is the line x = 4.

Reflecting flips the shape over the line, like a mirror.

Each point lands just as far away, on the other side.

reflection = flip

3 Reflect each point

From P, count straight to the mirror line: 2.

Then count 2 more on the other side.

We call the new point P prime.

The little mark means: the new P.

P is 2 left of the line.

Its image is 2 right of it.

Its y stays 3.

So P' is (6, 3).

P(2, 3) → P'(6, 3)

Careful!

Count to the mirror line x = 4, not to the y-axis.

Careful: not P'(−2, 3) ✗

4 The new point

So the image is P'(6, 3).

P' = (6, 3)

5 Check

Let's check: the mirror line is halfway between a point and its image.

Halfway between 2 and 6: add them, then divide by 2.

That is 4: the line x = 4.

Right!

P: 2 + 62 = 4

Let's look back.

First: draw the shape.

Then: draw the mirror line.

Then: reflect each point.

Then: the new point.

Last: check.

Answer
P' = (6, 3)

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. reflect (3, 2) in the x-axis
    Answer
    P' = (3, −2)
  2. reflect (3, 2) in the y-axis
    Answer
    P' = (−3, 2)
  3. reflect (-4, 1) in the x-axis
    Answer
    P' = (−4, −1)
  4. reflect A(1, 2) B(4, 2) in the y-axis
    Answer
    A'(−1, 2) and B'(−4, 2)
  5. reflect A(1, 1) B(4, 1) C(1, 3) in the x-axis
    Answer
    A'(1, −1), B'(4, −1) and C'(1, −3)
  6. reflect A(2, 1) B(4, 1) C(4, 3) in the y-axis
    Answer
    A'(−2, 1), B'(−4, 1) and C'(−4, 3)
  7. reflect (0, 3) in the x-axis
    Answer
    P' = (0, −3)
  8. reflect A(-3, 1) B(-1, 1) C(-1, 4) in the x-axis
    Answer
    A'(−3, −1), B'(−1, −1) and C'(−1, −4)

Medium

  1. reflect (1, 4) in the line y = x
    Answer
    P' = (4, 1)
  2. reflect A(1, 3) B(2, 5) C(4, 3) in the line y = x
    Answer
    A'(3, 1), B'(5, 2) and C'(3, 4)
  3. reflect A(-2, 1) B(-2, 4) C(1, 4) in the y-axis
    Answer
    A'(2, 1), B'(2, 4) and C'(−1, 4)
  4. reflect A(2, -1) B(5, -1) C(5, -3) in the x-axis
    Answer
    A'(2, 1), B'(5, 1) and C'(5, 3)
  5. reflect A(-2, 1) B(-1, 4) C(1, 3) in the line y = x
    Answer
    A'(1, −2), B'(4, −1) and C'(3, 1)
  6. reflect A(1, 2) B(3, 2) C(1, 4) over the y-axis
    Answer
    A'(−1, 2), B'(−3, 2) and C'(−1, 4)
  7. reflect A(3, 0) B(5, 2) C(3, 2) in the x-axis
    Answer
    A'(3, 0), B'(5, −2) and C'(3, −2)

Harder

  1. reflect (2, 3) in the line x = 4
    Answer
    P' = (6, 3)
  2. reflect A(1, 1) B(2, 4) C(3, 2) in the line x = 5
    Answer
    A'(9, 1), B'(8, 4) and C'(7, 2)
  3. reflect A(1, 1) B(4, 1) C(1, 3) in y = -1
    Answer
    A'(1, −3), B'(4, −3) and C'(1, −5)
  4. reflect A(-1, 1) B(1, 2) C(0, 4) in the line x = 2
    Answer
    A'(5, 1), B'(3, 2) and C'(4, 4)
  5. reflect A(2, 4) B(5, 4) C(5, 6) in the line y = 2
    Answer
    A'(2, 0), B'(5, 0) and C'(5, −2)
  6. reflect A(1, -1) B(3, -1) C(3, -4) in the line x = -1
    Answer
    A'(−3, −1), B'(−5, −1) and C'(−5, −4)