Sec 3 E-Math Congruence & Similarity topical questions

Triangles ABC and PQR satisfy AB = PQ, BC = QR and angle B = angle Q. Name the congruence test that applies.

A triangle has sides 4 cm, 5 cm and 6 cm. A similar triangle has shortest side 8 cm. Find the other two sides.

A figure with perimeter 12 cm and one side 2.5 cm is enlarged by scale factor 3. Find the new perimeter and the new length of that side.

On a map of scale 1 : 25 000, two towns are 6 cm apart. Find the real distance in kilometres.

A 1.6 m person casts a 2 m shadow. A flagpole casts a 7.5 m shadow at the same time. Find the height of the flagpole.

Triangles ABC and PQR are congruent, with A ↔ P, B ↔ Q and C ↔ R. Given AB = 5 cm, BC = 7 cm, angle B = 48° and angle C = 62°, state (a) the length PQ and (b) the angle R.

A model car is built to a scale of 1 : 18. The model is 24 cm long. Find the real length of the car in metres.

Triangles PQR and XYZ are similar with PQ = 6 cm, QR = 9 cm, PR = 12 cm and XY = 4 cm. (a) Find the scale factor taking ΔPQR to ΔXYZ. (b) Find YZ and XZ.

Similar triangles have one pair of corresponding sides 5 cm and 8 cm. Another pair measures (2x − 1) cm and (3x + 2) cm. Find x and state both lengths.

Two cubes have edges 3 cm and 2 cm. Find (a) the ratio of their total surface areas and (b) the ratio of their volumes, each in its simplest form.

Two similar cones have volumes 81 cm³ and 24 cm³. (a) Find the ratio of their heights. (b) The smaller cone is 4 cm tall; find the height of the larger cone.

Two similar triangles have areas 45 cm² and 20 cm². (a) Find the ratio of their corresponding sides. (b) A side of the smaller triangle is 6 cm; find the corresponding side of the larger triangle.

Two similar triangles have areas 50 cm² and 18 cm². (a) Find the ratio of their corresponding lengths. (b) The smaller triangle has perimeter 15 cm; find the perimeter of the larger. (c) A side of the larger triangle is 10 cm; find the corresponding side of the smaller.

1. (a)Length ratio.[2]
2. (b)Larger perimeter.[1]
3. (c)Smaller side.[2]

A vertical 1.5 m stick casts a shadow 1.2 m long. At the same moment a tower casts a 24 m shadow. (a) Explain why the stick-triangle and the tower-triangle are similar. (b) Find the height of the tower.

1. (a)Justify the similarity.[2]
2. (b)Find the tower height.[3]

A floor plan is drawn to a scale of 1 : 50. (a) A wall is 5 m long in reality. Find its length on the plan in centimetres. (b) A room has plan area 120 cm². Find its real area in square metres. (c) A 4 m × 3 m window is missing from the plan; compute its plan area.

1. (a)Wall length on plan.[2]
2. (b)Real area of the room.[2]
3. (c)Plan area of the window.[2]

In triangle ABC, D lies on AB and E lies on AC with DE parallel to BC. AD = (x + 2) cm, DB = (x − 1) cm, AE = 6 cm and EC = 4 cm. (a) Explain why triangle ADE is similar to triangle ABC. (b) Form an equation in x. (c) Find x and the length of AB.

1. (a)Justify the similarity.[2]
2. (b)Form the equation.[2]
3. (c)Solve and find AB.[2]

A model statue is built to a scale of 1 : 15 (model : real). (a) The model is 24 cm tall. Find the real height in metres. (b) The real statue’s arm span is 1.8 m. Find the arm span of the model in centimetres. (c) The model’s volume is 120 cm³. Find the real volume in cubic metres.

1. (a)Real height.[2]
2. (b)Model arm span.[2]
3. (c)Real volume.[2]

Two similar water tanks have heights 40 cm and 60 cm. (a) Find the ratio of their volumes. (b) The smaller tank holds 32 litres. Find the capacity of the larger tank. (c) The two tanks are repainted; the larger tank needs 1.8 litres of paint. How much paint does the smaller tank need?

1. (a)Volume ratio.[2]
2. (b)Larger capacity.[2]
3. (c)Paint for the smaller tank.[2]

In triangle ABC, D lies on AB and E lies on AC, with DE parallel to BC. AD = (2x − 1) cm, DB = (x + 4) cm, AE = 6 cm and EC = 9 cm. (a) Explain why triangle ADE is similar to triangle ABC. (b) Form an equation in x. (c) Find x, and state AD and AB.

1. (a)Justify the similarity.[2]
2. (b)Form the equation.[1]
3. (c)Solve and state AD, AB.[3]

A playground is drawn on a town-council plan at a scale of 1 : 500 as a rectangle 6 cm by 4.8 cm. (a) Find the real dimensions in metres. (b) Find the real area. (c) A football pitch of real area 3 600 m² is to be added to the same plan. What plan area would it occupy?

1. (a)Real dimensions.[2]
2. (b)Real area.[2]
3. (c)Plan area of the pitch.[2]