Geometry Banker

Four angles meet at a point. They are 4x°, (2x + 15)°, 95° and (x + 40)°.

1. (a) Find x. [2]
2. (b) Find the size of the largest angle. [1]

1. (a) Find the size of one exterior angle of a regular hexagon. [1]
2. (b) Find the size of one interior angle of a regular hexagon. [1]
3. (c) Find the sum of the interior angles of an octagon. [2]

1. (a) One angle of an isosceles triangle is 38° and it is the angle between the two equal sides. Find the other two angles. [2]
2. (b) In a different isosceles triangle, one of the two equal angles is 38°. Find the third angle. [1]

Each exterior angle of a regular polygon is 24°.

1. (a) Find the number of sides of the polygon. [2]
2. (b) Find the size of one interior angle. [1]
3. (c) Find the sum of the interior angles. [1]

AB and CD are parallel straight lines, with A and C on the left and B and D on the right. A straight line crosses AB at P and CD at Q, with Q below P. Angle BPQ = 64°.

1. (a) Find angle PQC. Give a geometrical reason. [2]
2. (b) Find angle PQD. Give a geometrical reason. [2]

Each interior angle of a regular polygon is 4 times as large as each exterior angle.

1. (a) Find the size of one exterior angle. [2]
2. (b) Find the number of sides of the polygon. [1]

The interior angles of a pentagon are x°, (x + 20)°, (2x − 10)°, 105° and (3x − 30)°.

1. (a) Find x. [3]
2. (b) Find the size of the largest angle. [1]

A square, a regular hexagon and a regular polygon with n sides meet at a point, fitting together with no gaps or overlaps.

1. (a) Find n. [3]

A, B and C are points on a circle. AB is a diameter. Angle CAB = 34°.

1. (a) Write down the size of angle ACB. Give a reason. [2]
2. (b) Find angle ABC. [1]

A, B and C are points on a circle with centre O. C is on the major arc AB. Angle AOB = 128°.

1. (a) Find angle ACB. Give a reason. [2]
2. (b) Find angle OAB. [2]

PQRS is a cyclic quadrilateral. Angle PQR = (2x + 10)° and angle PSR = (3x − 5)°.

1. (a) Find x. [2]
2. (b) Find angle PQR. [1]
3. (c) Angle QPS = 75°. Find angle QRS. [1]

A, B and C are points on a circle with centre O. TA is a tangent to the circle at A. Angle TAB = 58°, and C is on the major arc AB (on the opposite side of AB from T).

1. (a) Find angle ACB. Give a reason. [2]
2. (b) Angle CAB = 47°. Find angle ABC. [1]
3. (c) Find the reflex angle AOB. [2]

PA and PB are tangents from a point P to a circle with centre O. A and B are on the circle. Angle APB = 48°.

1. (a) Find angle AOB. [2]
2. (b) Find angle OAB. [1]
3. (c) C is a point on the major arc AB. Find angle ACB. [1]

Triangle ABC is similar to triangle PQR, with A corresponding to P, B to Q and C to R.AB = 6 cm, BC = 8 cm, AC = 9 cm and PQ = 9 cm.

1. (a) Find the scale factor of the enlargement from ABC to PQR. [1]
2. (b) Find QR and PR. [2]

1. (a) In triangles ABC and DEF, AB = DE, angle ABC = angle DEF and BC = EF. Which condition shows that the triangles are congruent? [1]
2. (b) ABCD is a parallelogram. Show that triangles ABC and CDA are congruent. [3]

Two bottles are mathematically similar. The smaller bottle is 12 cm tall and the larger bottle is 18 cm tall.

1. (a) The volume of the smaller bottle is 320 cm³. Find the volume of the larger bottle. [2]
2. (b) The surface area of the larger bottle is 405 cm². Find the surface area of the smaller bottle. [2]

In triangle ABC, D is a point on AB and E is a point on AC so that DE is parallel to BC.AD = 4 cm, DB = 6 cm and DE = 5 cm.

1. (a) Find BC. [2]
2. (b) The area of triangle ADE is 8 cm². Find the area of the trapezium DBCE. [3]

A map has a scale of 1 : 50 000.

1. (a) On the map, the distance between two villages is 7.4 cm. Find the actual distance in kilometres. [2]
2. (b) A road is 2.5 km long. How long is it on the map, in cm? [1]

1. (a) The bearing of B from A is 065°. Find the bearing of A from B. [1]
2. (b) The bearing of C from A is 310°. Find the bearing of A from C. [1]
3. (c) Find angle BAC, the angle between the directions of B and C from A. [1]

A plan of a rectangular garden is drawn to a scale of 1 : 200. On the plan the garden measures 9 cm by 6.5 cm.

1. (a) Find the actual length and width of the garden in metres. [2]
2. (b) Find the actual area of the garden. [1]
3. (c) A pond in the garden has an actual area of 6 m². Find its area on the plan in cm². [2]

A ship sails 8 km from port P on a bearing of 040° to a point Q. It then sails 6 km on a bearing of 130° to a point R.

1. (a) Find the bearing of P from Q. [1]
2. (b) Show that angle PQR = 90°. [2]
3. (c) Find the distance PR. [2]

1. (a) Write down the name of a quadrilateral that has exactly one pair of parallel sides. [1]
2. (b) Write down the name of a polygon with 7 sides. [1]
3. (c) Write down the type of each angle: 95°, 210°, 45°. [2]

1. (a) Write down the number of faces, edges and vertices of a triangular prism. [2]
2. (b) Write down the number of faces, edges and vertices of a square-based pyramid. [2]
3. (c) How many faces does the net of a cuboid have? [1]

1. (a) Write true or false for each statement.(i) The diagonals of a rhombus cross at right angles.(ii) The diagonals of a rectangle bisect its angles.(iii) A kite that is not a rhombus has exactly one line of symmetry.(iv) The opposite angles of a parallelogram are equal. [4]
2. (b) Name the quadrilateral whose diagonals are equal in length, bisect each other and cross at right angles. [1]

A is the point (−1, 1), B is (2, 5) and C is (7, 5).

1. (a) Show that AB = BC. [2]
2. (b) ABCD is a rhombus. Find the coordinates of D. [2]
3. (c) Find the coordinates of the point where the diagonals of the rhombus cross. [1]

1. (a) Write down the number of lines of symmetry and the order of rotational symmetry of each shape.(i) regular pentagon(ii) rectangle (not a square)(iii) parallelogram (not a rectangle or rhombus)(iv) isosceles trapezium [4]

Use the capital letters written in a plain block style: H, S, Z.

1. (a) For each capital letter H, S and Z, write down the number of lines of symmetry. [2]
2. (b) For each of the letters H, S and Z, write down the order of rotational symmetry. [2]
3. (c) Write down the order of rotational symmetry of a regular hexagon. [1]

1. (a) How many planes of symmetry does a cuboid measuring 2 cm by 3 cm by 5 cm have? [1]
2. (b) How many planes of symmetry does a right pyramid with a square base have? [1]
3. (c) A right prism has a cross-section that is an equilateral triangle. How many planes of symmetry does it have? [1]
4. (d) Write down the order of rotational symmetry of this triangular prism about the axis through the centres of its two triangular faces. [1]

1. (a) Describe the locus of points that are 3 cm from a fixed point A. [1]
2. (b) Describe the locus of points that are the same distance from two fixed points B and C. [1]
3. (c) Describe the locus of points that are the same distance from two straight lines that meet at a point. [1]
4. (d) A goat is tied to a post at one corner of a large rectangular field with a rope 5 m long. Describe the region the goat can reach. [1]

On a coordinate grid, A is the point (0, 0) and B is the point (8, 0).

1. (a) Find the equation of the locus of points that are the same distance from A and from B. [1]
2. (b) Find the coordinates of the two points on this locus that are 5 units from A. [2]
3. (c) Describe the region of points that are nearer to A than to B and less than 5 units from A. [2]