Similarity and congruence – Year 11 GCSE maths worksheet with answers to download as PDF

Two triangles are congruent if one of these conditions holds: SSS, SAS, ASA (or AAS) or RHS. Decide which condition, if any, proves each pair congruent.

1. Triangle 1: sides 5 cm, 7 cm, 9 cm. Triangle 2: sides 9 cm, 5 cm, 7 cm. [2 marks]
2. Triangle 1: sides 6 cm and 8 cm with an included angle of \(50^\circ\). Triangle 2: sides 6 cm and 8 cm with an included angle of \(50^\circ\). [2 marks]
3. Triangle 1 has angles \(40^\circ\) and \(60^\circ\) with 7 cm between them. Triangle 2 has angles \(40^\circ\) and \(60^\circ\) and a side of 7 cm between them. [2 marks]
4. Two right-angled triangles each have hypotenuse 13 cm and one side of 5 cm. [2 marks]
5. Triangle 1 has angles \(30^\circ\), \(60^\circ\), \(90^\circ\). Triangle 2 has angles \(30^\circ\), \(60^\circ\), \(90^\circ\) (no sides given). [2 marks]

A rectangle is 4 cm by 6 cm. A similar rectangle is 10 cm wide.

1. Find the scale factor of the enlargement. [2 marks]
2. Find the length of the larger rectangle. [2 marks]
3. Find the perimeter of each rectangle. [2 marks]
4. What is the ratio of the perimeters? Compare with the scale factor. [2 marks]

Triangle \(PQR\) has sides 6 cm, 8 cm and 10 cm. Triangle \(XYZ\) is similar with shortest side 9 cm.

1. Find the scale factor. [1 mark]
2. Find the other two sides of \(XYZ\). [3 marks]
3. Show that \(XYZ\) is right-angled. [2 marks]

In triangle \(ABC\), \(DE\) is parallel to \(BC\) with \(D\) on \(AB\) and \(E\) on \(AC\). \(AD=4\) cm, \(DB=2\) cm and \(DE=5\) cm.

1. Explain why triangles \(ADE\) and \(ABC\) are similar. [2 marks]
2. Find the scale factor from \(ADE\) to \(ABC\). [2 marks]
3. Find \(BC\). [2 marks]

Two similar solids have corresponding lengths in the ratio \(2:3\).

1. Write the ratio of their surface areas. [2 marks]
2. Write the ratio of their volumes. [2 marks]
3. The smaller has surface area 36 cm². Find the surface area of the larger. [2 marks]
4. The smaller has volume 40 cm³. Find the volume of the larger. [2 marks]

Triangle \(ABC\) is isosceles with \(AB=AC\). \(M\) is the midpoint of \(BC\).

1. Prove that triangles \(ABM\) and \(ACM\) are congruent. [3 marks]
2. Hence show that angle \(ABM=\) angle \(ACM\). [1 mark]
3. Show that \(AM\) is perpendicular to \(BC\). [3 marks]

A model car is built to a scale of \(1:24\). The real car is 4.32 m long.

1. Find the length of the model in centimetres. [2 marks]
2. The window on the model has area 25 cm². Find the area of the real window in m². [3 marks]
3. The real car has a boot of volume 400 litres. Find the volume of the model boot in cm³. [3 marks]

A photograph measures 15 cm by 10 cm. It is enlarged so that the width becomes 24 cm.

1. Find the scale factor. [1 mark]
2. Find the new height. [1 mark]
3. By what factor does the area increase? [2 marks]
4. The original photo is printed with 3 g of ink. Estimate the ink needed for the enlargement. [2 marks]

A person 1.7 m tall casts a shadow 2.4 m long. At the same time a tree casts a shadow 9.6 m long.

1. Explain why the triangles are similar. [2 marks]
2. Find the height of the tree. [3 marks]
3. A flagpole is 12.5 m tall. Find the length of its shadow at the same time. [2 marks]

A bronze statue is 3 m tall with a mass of 500 kg and a surface area of 12 m². A model of the statue is 30 cm tall and made from the same bronze.

1. Find the scale factor from statue to model. [1 mark]
2. Find the mass of the model. [3 marks]
3. Find the surface area of the model in cm². [3 marks]
4. The statue needs paint costing £4 per m². Work out the cost to paint the statue and the model. [3 marks]

Two similar cans have heights 8 cm and 12 cm. The smaller holds 250 ml and has a label of area 90 cm².

1. Find the length scale factor. [1 mark]
2. Find the volume of the larger can. [3 marks]
3. Find the area of the label on the larger can. [2 marks]
4. The larger can costs £1.40 and the smaller £0.80. Which is better value per ml? [3 marks]

On a map with scale \(1:50\,000\) a lake has an area of 3 cm².

1. Find the length that 1 cm on the map represents in km. [1 mark]
2. Find the real area of the lake in km². [4 marks]
3. A reservoir has real area 2 km². What is its area on the map? [2 marks]

Two similar swimming pools have depths 1.2 m and 1.8 m. The smaller has a surface area of 30 m² and a volume of 36 m³.

1. Find the length scale factor. [1 mark]
2. Find the surface area of the larger pool. [2 marks]
3. Find the volume of the larger pool. [2 marks]
4. Tiles for the floor cost £28 per m². Find the extra cost for the larger pool. [3 marks]

A pizza shop sells a 8-inch pizza for £6 and a 12-inch pizza for £11. The pizzas are circles with the given diameters and the same thickness.

1. Find the area of each pizza to 2 decimal places. [3 marks]
2. Show that the area ratio is \(4:9\). [2 marks]
3. Which pizza is better value? Show how you know. [3 marks]

\(ABCD\) is a parallelogram with diagonal \(AC\).

1. Prove that triangles \(ABC\) and \(CDA\) are congruent. [4 marks]
2. Hence prove that opposite angles \(B\) and \(D\) are equal. [2 marks]
3. Prove that the diagonals of a parallelogram bisect each other (use triangles \(AMB\) and \(CMD\) with \(M\) the intersection). [4 marks]

In triangle \(ABC\), \(DE\) is parallel to \(BC\). \(AD=6\) cm, \(DB=4\) cm, \(AE=9\) cm and \(BC=15\) cm.

1. Find \(EC\). [3 marks]
2. Find \(DE\). [3 marks]
3. Find the ratio of the area of triangle \(ADE\) to the area of trapezium \(DBCE\). [4 marks]

Two similar shapes have areas 20 cm² and 45 cm². Two similar solids have volumes 54 cm³ and 128 cm³.

1. Find the length scale factor for the shapes. [3 marks]
2. The smaller shape has a perimeter of 24 cm. Find the perimeter of the larger. [2 marks]
3. Find the length scale factor for the solids. [3 marks]
4. The smaller solid has surface area 36 cm². Find the surface area of the larger. [3 marks]

Two similar solids have volumes 125 cm³ and 216 cm³. The smaller solid has surface area 75 cm².

1. Find the length scale factor. [2 marks]
2. Find the surface area of the larger solid. [3 marks]
3. Write the ratio of the volumes in its simplest form. [1 mark]

\(ABCD\) is a kite with \(AB=AD\) and \(CB=CD\).

1. Prove that triangles \(ABC\) and \(ADC\) are congruent. [3 marks]
2. Prove that \(AC\) bisects angle \(BAD\). [2 marks]
3. Prove that \(AC\) bisects angle \(BCD\). [1 mark]
4. Prove that \(AC\) is perpendicular to \(BD\) (hint: use the point where they meet). [4 marks]

Triangle \(ABC\) has a right angle at \(A\) with \(AB=6\) cm and \(AC=8\) cm. \(AD\) is perpendicular to \(BC\) with \(D\) on \(BC\).

1. Find \(BC\). [1 mark]
2. Prove that triangles \(ABD\) and \(CBA\) are similar and find \(BD\). [5 marks]
3. Find \(AD\) and \(DC\). [3 marks]
4. Show that \(AD^2=BD\times DC\). [2 marks]