Teach Me Maths

Reflections

Question 11 mark

Reflect point A(3,2) in the y-axis. Give the coordinates of its image.

Write your answer as (x, y)

Hint

Count how far A is from the vertical y-axis. Keep its height the same.

Worked solution

Reflect point A(3,2) in the y-axis. Give the coordinates of its image.

  1. A is 3 units right of the y-axis. Its reflection is 3 units left of the y-axis.
  2. The y-coordinate stays 2, so the image is (-3,2).

Question 21 mark

Reflect B(-2,4) in the x-axis. Give the coordinates of its image.

Write your answer as (x, y)

Hint

The x-axis is horizontal. Keep the same x-coordinate.

Worked solution

Reflect B(-2,4) in the x-axis. Give the coordinates of its image.

  1. B is 4 units above the x-axis, so its image is 4 units below it.
  2. The x-coordinate stays -2, giving (-2,-4).

Question 32 marks

Reflect C(1,3) in the line x=2. Give the coordinates of its image.

Write your answer as (x, y)

Hint

The mirror is the vertical line through x = 2.

Worked solution

Reflect C(1,3) in the line x=2. Give the coordinates of its image.

  1. C is 1 unit left of x=2. Its image is 1 unit right of that line, at x=3.
  2. The y-coordinate stays 3, so the image is (3,3).

Question 42 marks

Reflect D(4,-1) in the line y=1. Give the coordinates of its image.

Write your answer as (x, y)

Hint

Count the vertical distance from −1 up to the mirror at y = 1.

Worked solution

Reflect D(4,-1) in the line y=1. Give the coordinates of its image.

  1. D is 2 units below y=1. Its image is 2 units above the mirror, at y=3.
  2. The x-coordinate stays 4, so the image is (4,3).

Question 51 mark

Point E is (2,-3). It is reflected in the line x=2. What are the coordinates of its image?

Write your answer as (x, y)

Hint

A point exactly on the mirror does not move.

Worked solution

Point E is (2,-3). It is reflected in the line x=2. What are the coordinates of its image?

  1. E lies on x=2, so its perpendicular distance from the mirror is zero.
  2. It stays at (2,-3).

Question 63 marks

Reflect triangle ABC in the y-axis. Give the coordinates of its image vertices. Each checked vertex will appear on the grid.

(a)

Find A′.

1 mark

Write your answer as (x, y)

(b)

Find B′.

1 mark

Write your answer as (x, y)

(c)

Find C′.

1 mark

Write your answer as (x, y)

Hint

Reflect each vertex separately. Keep each vertex label with its own image.

Worked solution

Reflect triangle ABC in the y-axis. Give the coordinates of its image vertices. Each checked vertex will appear on the grid.

Part (a)

Find A′.

  1. The y-coordinate stays the same and the x-coordinate changes sign.
  2. A(-4, 1) maps to A′(4, 1).

Part (b)

Find B′.

  1. The y-coordinate stays the same and the x-coordinate changes sign.
  2. B(-1, 1) maps to B′(1, 1).

Part (c)

Find C′.

  1. The y-coordinate stays the same and the x-coordinate changes sign.
  2. C(-4, 3) maps to C′(4, 3).
  3. The completed image is shown below.
    Reflection on a coordinate gridA coordinate grid with one-unit grid intervals and equal scales on both axes. A at (-4, 1), B at (-1, 1), C at (-4, 3), A′ at (4, 1), B′ at (1, 1), C′ at (4, 3). The vertices of ABC, A′B′C′ are joined in order. The dashed mirror line is x = 0.Mirror: x = 0−5−4−3−2−112345−5−4−3−2−112345xyOABCA′B′C′
    The image vertices are joined in the same order to give the reflected triangle.

Question 72 marks

Reflect G(3,1) in the line y=x. Give the image coordinates.

Write your answer as (x, y)

Hint

For this diagonal mirror, the x- and y-coordinates exchange places.

Worked solution

Reflect G(3,1) in the line y=x. Give the image coordinates.

  1. Draw the perpendicular from G(3,1) to the mirror. It reaches M(2,2): 1 square left and 1 up.
    Perpendicular reflection constructionG at (3, 1) reflects to (1, 3). The join meets the mirror at midpoint (2.0, 2.0) at a right angle. Matching ticks mark equal distances on the two halves. Every grid square has equal horizontal and vertical scale.-1-111223344550xyMGG′Mirror: y = xMatching ticks: equal distances
    M is the midpoint. The matching ticks show equal distances from the mirror.
  2. Continue the same distance beyond M: another 1 left and 1 up gives G′(1,3).
  3. The mirror y=x lies halfway between the horizontal and vertical directions. Reflecting across it exchanges the horizontal and vertical positions, so (x,y)\mapsto(y,x). Here 3 and 1 exchange places.

Question 82 marks

Reflect H(-3,2) in the line y=x. Give the image coordinates.

Write your answer as (x, y)

Hint

Swap the coordinates, carrying each sign with its number.

Worked solution

Reflect H(-3,2) in the line y=x. Give the image coordinates.

  1. Draw the perpendicular from H(-3,2) to the mirror. It reaches M(-0.5,-0.5): 2.5 squares right and 2.5 down.
    Perpendicular reflection constructionH at (-3, 2) reflects to (2, -3). The join meets the mirror at midpoint (-0.5, -0.5) at a right angle. Matching ticks mark equal distances on the two halves. Every grid square has equal horizontal and vertical scale.-5-5-4-4-3-3-2-2-1-111223344550xyMHH′Mirror: y = xMatching ticks: equal distances
    The right-angle mark and equal half-distance ticks show the reflection construction.
  2. Continue another 2.5 squares right and 2.5 down to H′(2,-3).
  3. Across y=x, horizontal and vertical positions exchange. Thus (x,y)\mapsto(y,x): the original y-coordinate 2 becomes the new x-coordinate, and the original x-coordinate -3 becomes the new y-coordinate.

Question 92 marks

Reflect J(2,4) in the line y=-x. Give the image coordinates.

Write your answer as (x, y)

Hint

Swap the coordinates and change both signs.

Worked solution

Reflect J(2,4) in the line y=-x. Give the image coordinates.

  1. Draw a perpendicular from J to the mirror. It meets the mirror at M(-1,1): move 3 squares left and 3 down from J(2,4).
    Perpendicular reflection constructionJ at (2, 4) reflects to (-4, -2). The join meets the mirror at midpoint (-1.0, 1.0) at a right angle. Matching ticks mark equal distances on the two halves. Every grid square has equal horizontal and vertical scale.-5-5-4-4-3-3-2-2-1-111223344550xyMJJ′Mirror: y = −xMatching ticks: equal distances
    The join is at right angles to the mirror. M is halfway between J and J′.
  2. Continue the same distance beyond M: another 3 left and 3 down gives J′(-4,-2).
  3. For this diagonal, the coordinates exchange places and both signs change: (x,y)\mapsto(-y,-x). This gives (2,4)\mapsto(-4,-2).

Question 101 mark

Which transformation maps triangle ABC onto triangle A′B′C′?

Hint

Compare corresponding vertices, including B and B′.

Worked solution

Which transformation maps triangle ABC onto triangle A′B′C′?

  1. Reflection in the y-axis.
  2. Translation four units right.
  3. Reflection in the line x = 1.
  4. Reflection in the line y = 1.
  1. Each pair of matching vertices has the same y-coordinate.
  2. Their x-coordinates have midpoint 1, so the mirror is x=1.
  3. The transformation is reflection in the line x=1.

Question 113 marks

Reflect triangle ABC in the line y=-1. Give the coordinates of its image vertices. Each checked vertex will appear on the grid.

(a)

Find A′.

1 mark

Write your answer as (x, y)

(b)

Find B′.

1 mark

Write your answer as (x, y)

(c)

Find C′.

1 mark

Write your answer as (x, y)

Hint

Reflect each vertex separately. Keep each vertex label with its own image.

Worked solution

Reflect triangle ABC in the line y=-1. Give the coordinates of its image vertices. Each checked vertex will appear on the grid.

Part (a)

Find A′.

  1. Keep the x-coordinate. The image is the same vertical distance on the other side of y=-1.
  2. A(1, 1) maps to A′(1, -3).

Part (b)

Find B′.

  1. Keep the x-coordinate. The image is the same vertical distance on the other side of y=-1.
  2. B(4, 1) maps to B′(4, -3).

Part (c)

Find C′.

  1. Keep the x-coordinate. The image is the same vertical distance on the other side of y=-1.
  2. C(1, 3) maps to C′(1, -5).
  3. The completed image is shown below.
    Reflection on a coordinate gridA coordinate grid with one-unit grid intervals and equal scales on both axes. A at (1, 1), B at (4, 1), C at (1, 3), A′ at (1, -3), B′ at (4, -3), C′ at (1, -5). The vertices of ABC, A′B′C′ are joined in order. The dashed mirror line is y = -1.Mirror: y = -1−5−4−3−2−112345−6−5−4−3−2−112345xyOABCA′B′C′
    The image vertices are joined in the same order to give the reflected triangle.

Question 122 marks

Triangle PQR is reflected to triangle P′Q′R′. The mirror line is x=k. Find k.

Hint

The vertical mirror is halfway between the two x-coordinates.

Worked solution

Triangle PQR is reflected to triangle P′Q′R′. The mirror line is x=k. Find k.

  1. The x-coordinates are -3 and 5, a distance of 8 units apart.
  2. Halfway is 4 units right of -3: -3+4=1.
  3. The mirror line is x=1, so k=1.

Question 132 marks

Triangle PQR is reflected to triangle P′Q′R′. The mirror line is y=k. Find k.

Hint

Use the midpoint of the y-coordinates because the mirror is horizontal.

Worked solution

Triangle PQR is reflected to triangle P′Q′R′. The mirror line is y=k. Find k.

  1. For P and P′, the y-coordinates are -4 and 2. Their midpoint is (-4+2)\div2=-1.
  2. The other matching pairs have the same y-midpoint, so the mirror line is y=-1 and k=-1.

Question 141 mark

Mia was asked to reflect triangle ABC in x=1. Her image A′B′C′ is shown. Which statement describes her mistake?

Hint

Find the line halfway between each original vertex and its claimed image.

Worked solution

Mia was asked to reflect triangle ABC in x=1. Her image A′B′C′ is shown. Which statement describes her mistake?

  1. She reflected in the y-axis.
  2. She reflected in y = 1 instead of x = 1.
  3. She moved every vertex two units down.
  4. She rotated the triangle through a half-turn.
  1. A(-2,3) is shown at A′(-2,-1); their midpoint has y=1.
  2. B and C also keep their x-coordinates and reflect across y=1.
  3. Mia used the horizontal line y=1 instead of the vertical line x=1.

Question 153 marks

Reflect triangle ABC in the line y=x. Give the coordinates of its image vertices. Each checked vertex will appear on the grid.

(a)

Find A′.

1 mark

Write your answer as (x, y)

(b)

Find B′.

1 mark

Write your answer as (x, y)

(c)

Find C′.

1 mark

Write your answer as (x, y)

Hint

Reflect each vertex separately. Keep each vertex label with its own image.

Worked solution

Reflect triangle ABC in the line y=x. Give the coordinates of its image vertices. Each checked vertex will appear on the grid.

Part (a)

Find A′.

  1. Draw a perpendicular from A(-2,1) to the mirror. It reaches M(-0.5,-0.5): 1.5 squares right and 1.5 down.
    Perpendicular reflection constructionA at (-2, 1) reflects to (1, -2). The join meets the mirror at midpoint (-0.5, -0.5) at a right angle. Matching ticks mark equal distances on the two halves. Every grid square has equal horizontal and vertical scale.-5-5-4-4-3-3-2-2-1-111223344550xyMAA′Mirror: y = xMatching ticks: equal distances
    The right angle and matching ticks show that A and A′ are equally far from the mirror.
  2. Continue another 1.5 squares right and 1.5 down to A′(1,-2). Across y=x, horizontal and vertical positions exchange, giving (x,y)\mapsto(y,x).

Part (b)

Find B′.

  1. Reflect B(-4,1) along a perpendicular to y=x. Its midpoint on the mirror is M(-1.5,-1.5): 2.5 squares right and 2.5 down.
  2. Continue the same distance beyond the mirror: another 2.5 right and 2.5 down gives B′(1,-4). This exchanges B’s horizontal and vertical positions.

Part (c)

Find C′.

  1. Reflect C(-2,3) along a perpendicular to y=x. Its midpoint on the mirror is M(0.5,0.5): 2.5 squares right and 2.5 down.
  2. Continue another 2.5 squares right and 2.5 down to C′(3,-2).
  3. Join A′, B′ and C′ in the same order to form the reflected triangle.
    Perpendicular reflection constructionA at (-2, 1) reflects to (1, -2). The join meets the mirror at midpoint (-0.5, -0.5) at a right angle. Matching ticks mark equal distances on the two halves. Every grid square has equal horizontal and vertical scale.-5-5-4-4-3-3-2-2-1-111223344550xyABCA′B′C′MMirror: y = xMatching ticks: equal distances
    The A-to-A′ construction remains visible; the same equal-distance method gives B′ and C′.

Question 16Challenge3 marks

A reflection maps P(4,-2) to P′(-2,-2). The same reflection maps Q(-3,4) to Q′. Find Q′.

Write your answer as (x, y)

Hint

Find the mirror line halfway between P and P′, then reflect Q in that line.

Worked solution

A reflection maps P(4,-2) to P′(-2,-2). The same reflection maps Q(-3,4) to Q′. Find Q′.

  1. P and P′ have the same y-coordinate. The vertical mirror is halfway between x=4 and x=-2.
  2. The midpoint is (4-2)\div2=1, so the mirror is x=1.
  3. Q is 4 units left of the mirror, so Q′ is 4 units right: (5,4).

Question 17Challenge3 marks

A point is reflected in the line y=-1. Its image is (3,4). Find the original point.

Write your answer as (x, y)

Hint

A reflection works backwards in the same way. Count the distance from the image to the mirror.

Worked solution

A point is reflected in the line y=-1. Its image is (3,4). Find the original point.

  1. The image is 4-(-1)=5 units above the mirror.
  2. The original is 5 units below the mirror, at y=-1-5=-6.
  3. The x-coordinate stays 3, so the original was (3,-6).

Question 18Challenge2 marks

Triangle ABC is reflected to triangle A′B′C′. Write the equation of the mirror line.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Check that the same mirror line lies halfway between all matching vertex pairs.

Worked solution

Triangle ABC is reflected to triangle A′B′C′. Write the equation of the mirror line.

  1. A and A′ have x-midpoint (1-3)\div2=-1. B and B′ have x-midpoint (3-5)\div2=-1. C gives the same midpoint.
  2. All paired y-coordinates stay the same. The single transformation is reflection in x=-1.
  3. A translation would move every point by the same amount; the x-movements here are -4 and -8, so it is not a translation.

Question 19Challenge3 marks

A single reflection maps A(-2,-3) to A′(3,2) and maps B(4,-2) to B′(2,-4). Write the equation of the mirror line.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Each point and its image are the same distance from the mirror line, so the mirror line passes through the midpoint of AA′ and the midpoint of BB′. Sketching the points on a grid helps.

Worked solution

A single reflection maps A(-2,-3) to A′(3,2) and maps B(4,-2) to B′(2,-4). Write the equation of the mirror line.

  1. Midpoint of AA′: \left(\frac{-2+3}{2},\frac{-3+2}{2}\right)=(0.5,-0.5).
  2. Midpoint of BB′: \left(\frac{4+2}{2},\frac{-2+(-4)}{2}\right)=(3,-3).
  3. At both midpoints the y-coordinate is the negative of the x-coordinate, so the mirror line is the diagonal line y=-x.
  4. Check: reflecting in y=-x swaps the coordinates and changes both signs, so (-2,-3)\to(3,2) and (4,-2)\to(2,-4).

Question 20Challenge3 marks

A learner reflected triangle ABC in x=1. Their image A′B′C′ is shown. Exactly one image vertex is wrong. Identify it and give its correct coordinates.

(a)

Which image vertex is wrong?

1 mark

(b)

Give the correct coordinates for that image.

2 marks

Write your answer as (x, y)

Hint

Check each vertex’s perpendicular distance from the stated mirror line.

Worked solution

A learner reflected triangle ABC in x=1. Their image A′B′C′ is shown. Exactly one image vertex is wrong. Identify it and give its correct coordinates.

Part (a)

Which image vertex is wrong?

  1. A′
  2. B′
  3. C′
  1. A and C are 4 units left of x=1, so their images have x=5, as shown.
  2. B is 2 units left of the mirror. The claimed B′ lies on the mirror, so B′ is wrong.

Part (b)

Give the correct coordinates for that image.

  1. B is 1-(-1)=2 units left of x=1.
  2. Its image is 2 units right of the mirror, at x=1+2=3. Its y-coordinate stays -2.
  3. The correct image is B′(3,-2).
  4. The completed image is shown below.
    Reflection on a coordinate gridA coordinate grid with one-unit grid intervals and equal scales on both axes. A at (-3, -2), B at (-1, -2), C at (-3, 2), A′ at (5, -2), B′ at (3, -2), C′ at (5, 2). The vertices of ABC, A′B′C′ are joined in order. The dashed mirror line is x = 1.Mirror: x = 1−5−4−3−2−1123456−4−3−2−11234xyOABCA′B′C′
    This is the corrected reflected triangle.

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