GCSE Maths Congruence & transforms questions by topic

Naming congruence.

1. (a)State what it means for two shapes to be congruent.[1]
2. (b)Triangles with sides 3 cm, 4 cm, 5 cm and 3 cm, 4 cm, 5 cm are drawn. Which criterion proves them congruent?[1]
3. (c)Two triangles have angles 52° and 61° with a 7 cm side between them, each. Which criterion applies, and why does the side have to sit between the angles?[2]

Similar rectangles.

1. (a)A rectangle measures 4 cm by 6 cm. A similar rectangle has a longer side of 9 cm. Find the length scale factor.[2]
2. (b)Find the shorter side of the larger rectangle.[2]

Moving points on a grid.

1. (a)Translate the point (2, 3) by the vector (−3, −1). Give the new coordinates.[2]
2. (b)Reflect the point (5, 2) in the y-axis. Give the new coordinates.[2]

Enlargement lengths.

1. (a)A 3 cm segment is enlarged by scale factor 4. How long is the image?[2]
2. (b)A different segment is enlarged by scale factor ½ and its image is 4 cm. How long was the original?[2]

Two triangles are matched side for side.

1. (a)Triangle A has sides 6 cm, 8 cm, 10 cm; triangle B has sides 8 cm, 10 cm, 6 cm. State a congruence criterion proving A and B congruent.[2]
2. (b)Triangle C has two sides of 6 cm and 8 cm with the right angle between them. Explain why C is congruent to A.[2]

A grid point is reflected, then translated.

1. (a)Reflect the point (2, 5) in the y-axis.[2]
2. (b)Translate the point (2, 5) by the vector (3, −2).[2]

Similar triangles.

1. (a)Triangle A has sides 5 cm, 7 cm and 9 cm. Triangle B is similar to A with longest side 27 cm. Find the scale factor.[2]
2. (b)Find the other two sides of B.[2]
3. (c)State the relationship between the angles of A and the angles of B.[1]

Rotations about the origin.

1. (a)Rotate the point (3, 1) through 90° clockwise about the origin. Give the image.[2]
2. (b)Rotate the point (0, 4) through 180° about the origin. Give the image.[2]
3. (c)Which point is invariant under the rotation in part (b)?[1]

A photo enlargement.

1. (a)A 6 cm × 8 cm photo is enlarged so the longer side becomes 20 cm. Find the scale factor.[2]
2. (b)Find the shorter side of the enlarged photo.[1]
3. (c)The original photo has perimeter 28 cm. Find the perimeter of the enlargement.[2]

Vector arithmetic.

1. (a)Work out (4, −2) + (−1, 5).[2]
2. (b)Work out 2 × (−3, 4).[2]
3. (c)Write the translation “4 left, 1 up” as a column vector.[1]

A rectangle photo is blown up.

1. (a)A 4 cm by 10 cm photo is enlarged to 10 cm by 25 cm. Work out the length scale factor.[1]
2. (b)Write down the perimeter of each rectangle and the perimeter scale factor.[2]
3. (c)Write down the area of each rectangle and the area scale factor.[2]

Area and volume scale factors. (Higher)

1. (a)Two similar prisms have length scale factor 3. State their area scale factor and volume scale factor.[2]
2. (b)The smaller prism has volume 12 cm³. Find the volume of the larger.[2]
3. (c)The larger prism has surface area 117 cm². Find the surface area of the smaller.[2]

Negative enlargement and invariance. (Higher)

1. (a)An enlargement has centre O and scale factor −2. Find the image of the point (2, 0) and describe where it lies.[3]
2. (b)State the invariant points of this enlargement.[1]
3. (c)Explain what the negative scale factor does to the position and size of the image.[2]

Diagonals of a parallelogram. (Higher)

1. (a)OACB is a parallelogram with OA = a and OB = b. Write AB in terms of a and b.[2]
2. (b)M is the midpoint of AB. Show that OM = ½(a + b).[2]
3. (c)Hence explain why the diagonals of a parallelogram bisect each other.[2]

Combining transformations. (Higher)

1. (a)Reflect the point (2, 5) in the line y = x.[2]
2. (b)Translate your answer by the vector (−1, −3). Give the final point.[2]
3. (c)A shape is reflected in the x-axis and then in the y-axis. Describe the single transformation equivalent to the combination.[2]

A rotated triangle is described fully. (Higher)

1. (a)A triangle with vertices (1, 1), (4, 1) and (4, 3) is mapped to (−1, 1), (−1, 4) and (−3, 4). Give a full description of the transformation.[3]
2. (b)State the invariant points of the transformation.[2]

Two water tanks are similar. (Higher)

1. (a)Two similar tanks have surface areas 250 cm² and 640 cm². Work out the length scale factor from small to large.[3]
2. (b)State the factor by which volumes are multiplied.[1]
3. (c)The small tank holds 5 litres. Work out the capacity of the large tank.[2]

Diagonals of a parallelogram compared. (Higher)

1. (a)OACB is a parallelogram with OA = a and OB = b. Write AC and OC as vectors in terms of a and b.[2]
2. (b)M is the midpoint of OC. Write the position vector of M in terms of a and b.[2]
3. (c)N is the midpoint of AB. Show that N has the same position vector as M, and state what this proves about the diagonals.[2]