Sec 2 Mathematics Congruence & Similarity topical questions

Decide whether each pair is congruent, with a one-line reason.

1. (a)Two equilateral triangles, each with sides of 4 cm.[1]
2. (b)Two rectangles measuring 6 cm × 2 cm and 4 cm × 3 cm.[1]
3. (c)Two circles, each of radius 5 cm.[1]

A 4 cm by 6 cm rectangle is compared with a 10 cm by 15 cm rectangle.

1. (a)Find the scale factor from the small rectangle to the large one.[1]
2. (b)Confirm that the other side matches the same scale factor.[1]
3. (c)State what is true about the corresponding angles of the two rectangles.[1]

A triangle with sides 2 cm, 5 cm and 7 cm is enlarged by scale factor 3.

1. (a)Find the shortest side of the image.[1]
2. (b)Find the second side of the image.[1]
3. (c)Find the longest side of the image.[1]

A triangle with sides 3 cm, 8 cm and 10 cm is enlarged by scale factor 2.5.

1. (a)Find the image of the 3 cm side.[1]
2. (b)Find the image of the 8 cm side.[1]
3. (c)Find the image of the 10 cm side.[1]

Decide whether each pair must be congruent, with a one-line reason.

1. (a)Two circles, each of radius 3 cm.[1]
2. (b)Two squares, one of side 5 cm and one of side 6 cm.[1]
3. (c)Two rectangles both measuring 2 cm by 8 cm, one drawn rotated.[1]

A hiking map has scale 1 : 25 000.

1. (a)Two gates are 6 cm apart on the map. Find the real distance in kilometres.[2]
2. (b)A stream runs 3.5 km in reality. How long is it on the map, in centimetres?[2]

Triangle T has sides 4 cm, 9 cm and 11 cm. Triangle U is similar to T, and its shortest side is 12 cm.

1. (a)Find the scale factor from T to U.[1]
2. (b)Find the other two sides of U.[2]
3. (c)Find the perimeter of U.[1]

A photograph measures 8 cm by 12 cm and is enlarged by scale factor 2.5.

1. (a)Find the dimensions of the enlarged photograph.[2]
2. (b)Will the enlargement fit inside a frame 25 cm by 32 cm? Justify your answer.[2]

Triangle P has sides 2 cm, 3.5 cm and 5 cm. Triangle Q is similar to P with shortest side 7 cm.

1. (a)An angle in P measures 48°. What is the matching angle in Q?[1]
2. (b)Find the scale factor from P to Q.[1]
3. (c)Find the two remaining sides of Q.[2]

Triangle S has sides 4 cm, 6 cm and 9 cm. Triangle L is similar to S, and the perimeter of L is 76 cm.

1. (a)Find the perimeter of S.[1]
2. (b)Find the scale factor from S to L.[1]
3. (c)Find the three sides of L.[2]

A street directory uses a scale of 1 : 150 000.

1. (a)Two bus stops are 8 cm apart on the map. Find the real distance in kilometres.[2]
2. (b)An expressway runs 7.5 km in reality. How long is it on the map?[1]
3. (c)A third landmark lies 12 km from the bus stops. How far apart are they on the map?[1]

Triangles P, Q and R are linked: P is similar to Q, and Q is similar to R. P has sides 3 cm, 4 cm and 6 cm. Q’s longest side is 12 cm, and R’s shortest side is 15 cm.

1. (a)Find the scale factor from P to Q.[1]
2. (b)Write down the three sides of Q.[1]
3. (c)Find the scale factor from Q to R.[1]
4. (d)Find the longest side of R.[1]

On map A, drawn to scale 1 : 50 000, two towns lie 7.2 cm apart. On map B the same two towns lie 4.5 cm apart.

1. (a)Find the real distance between the towns, in kilometres.[2]
2. (b)Find the scale of map B in the form 1 : n.[3]

A model lorry is built at scale 1 : 24.

1. (a)The model is 35 cm long. Find the real length in metres.[2]
2. (b)The real lorry is 3 m tall. Find the model’s height in centimetres.[2]
3. (c)A wheel on the model has diameter 4 cm. Find the real diameter in centimetres.[1]

Triangle A is right-angled with sides 5 cm, 12 cm and 13 cm. Triangle B is similar to A with shortest side 15 cm. Triangle C is similar to B with longest side 62.4 cm.

1. (a)Find the scale factor from A to B.[1]
2. (b)Write down the three sides of B.[1]
3. (c)Find the scale factor from B to C.[1]
4. (d)State the three sides of C and whether C is right-angled.[2]