Printable · GCSE Foundation · ages 14-16
Geometry and measures worksheet — GCSE Foundation
Fifteen questions across the geometry and measures statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
Geometry and measures worksheet — GCSE Foundation
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- 1.A straight line crosses two parallel lines. One of the angles formed is (2x + 10)°, and the angle alternate to it is 74°. Work out the value of x.
- 2.Priya says: 'To find the arc length of a sector, you divide the circumference by the angle in degrees.' Which statement about Priya's method is correct?
- 3.A landscaper marks out two triangular flower beds using stakes and a tape measure. Bed 1 has two edges of 5 m and 7 m, meeting at a corner with a marked angle of 60° between them. Bed 2 has edges of 5 m and 7 m, meeting at a corner also marked at 60° between them. The landscaper wants to check the two beds will be exactly the same size and shape using only these three measurements. Which condition proves the two beds are congruent?
- 4.A cyclist rides around a perfectly circular track with centre O. At every moment of the ride, the distance from O to the cyclist is exactly the same. Which term names this fixed distance?
- 5.A ladder rests against a wall, making an angle of 60° with the ground. The ladder is 2 m long. Using the exact value of sin 60°, work out how high up the wall the ladder reaches, giving your answer in centimetres.
- 6.A rectangle has vertices at (2, 1), (9, 1), (9, 5) and (2, 5). Work out the area of the rectangle.
- 7.A scale model of a bridge is built so that 2 cm on the model represents 5 m of the real bridge. The real bridge is 45 m long. Work out the length of the model bridge, in centimetres.
- 8.Triangle ABC has AB = 8 cm, BC = 10 cm and CA = 6 cm. Triangle LMN has LM = 10 cm, MN = 6 cm and NL = 8 cm. The two triangles are congruent by SSS. Which of the following correctly matches the corresponding vertices?
- 9.Put sin 30°, tan 30° and cos 30° in order of size, starting with the smallest.
- 10.A right-angled triangle has angles of 90°, 40° and x°. Work out the value of x.
- 11.In quadrilateral WXYZ, which of the following correctly names the angle at vertex Y using standard notation?
- 12.A builder lays a low wall made from identical bricks. The diagram shows the plan of the wall's footprint on a grid, where each square is one brick. The wall is built up to a height of 4 bricks everywhere. Work out the total number of bricks used.
- 13.A fan-shaped rug is in the shape of a sector of a circle with radius 4 m and angle 180°. It costs £4 per square metre to make. Using π = 3.14, work out the total cost of the rug.
- 14.u is the column vector with top number 7 and bottom number 3. v is the column vector with top number 2 and bottom number 5. Work out u − v, giving your answer as a column vector in the form (top, bottom).
- 15.A hiker walks on a bearing of 065°. She then turns clockwise through 90° and continues walking in a straight line. What bearing is she now walking on?
Answer key
- (c) 32 — Method: alternate angles between parallel lines are equal, so 2x + 10 = 74. Working: subtracting 10 from both sides gives 2x = 64; dividing by 2 gives x = 32. Answer: x = 32. A candidate who forgets to subtract 10 first and divides 74 by 2 directly gets 37. A candidate who treats the angles as co-interior instead of alternate, so that the two expressions add to 180° rather than being equal, gets 48 after solving. A candidate who makes a sign error and treats the equation as 2x equalling 10 minus 74 instead of 74 minus 10 gets −32.
- (a) Wrong: multiply by angle ÷ 360, not divide by it. — The arc length is the fraction angle ÷ 360 of the circumference, and that fraction is MULTIPLIED by the circumference — it is never divided by the angle itself. So Priya's method, dividing by the angle, is wrong, and the correct statement names both the fraction and the multiplication. Saying she is correct accepts her wrong division. Saying the arc length is the full circumference multiplied by the angle in degrees, with no division at all, drops the 360 that turns the angle into a fraction. Saying the arc length is the radius multiplied by the angle in degrees swaps the circumference for the radius, which is a different and, in degrees, incorrect formula.
- (b) SAS — Method: check which condition matches two sides and the angle between them, since that is what has been measured for each bed. Working: the 60° angle is marked at the corner where the 5 m and 7 m edges meet, so it is the included angle — this is two Sides and the included Angle, SAS. Options: SSS would need a third side measured, but only two edges are known; ASA would need two angles and the side between them, but only one angle is measured; RHS needs a right angle, and 60° is not a right angle. Answer: SAS.
- (a) radius — The distance from the centre O to any point on the circle is the radius. The diameter is the distance right across the circle, through the centre, from one side to the other — twice the radius, not the same measurement. The circumference is the total distance around the circle, not from the centre. A chord is a line joining two points on the circle that does not have to pass through the centre at all.
- (c) 100√3 cm — The height is opposite the 60° angle, so height = 2 × sin 60° = 2 × √3/2 = √3 m. Converting to centimetres: √3 m = 100√3 cm. '√3 cm' forgets to convert the answer from metres to centimetres. '200√3 cm' comes from mis-recalling sin 60° as √3 instead of √3/2, dropping the denominator of the exact value: 2 × √3 = 2√3 m = 200√3 cm. '50√3 cm' comes from halving the ladder's length before multiplying by sin 60°, instead of using the full 2 m.
- (b) 28 — The width of the rectangle is the difference in x-coordinates, 9 − 2 = 7, and the height is the difference in y-coordinates, 5 − 1 = 4. The area is width × height = 7 × 4 = 28. 22 comes from using the perimeter formula, 2 × (7 + 4), instead of the area formula. 35 comes from multiplying 7 by 5 instead of 4, misreading one of the y-coordinates. 63 comes from multiplying 9 by 7, using an x-coordinate instead of the height.
- (c) 18 — Method: divide the real length by 5 to find how many 'units' of 5 m it contains, then multiply by 2 cm for each unit. Working: 45 ÷ 5 = 9, so the real bridge is 9 lots of 5 m; each lot is represented by 2 cm on the model, so the model length is 9 × 2 = 18 cm. Options: 9 comes from stopping after the division, without multiplying by the 2 cm per unit; 90 comes from multiplying the real length by 2 directly, without dividing by 5 first; 4.5 comes from dividing by 5 and then dividing by 2 again, instead of multiplying by 2. Answer: 18.
- (c) A↔N, B↔L, C↔M (ABC≅NLM) — Matching equal side lengths: AB (8 cm) equals NL (8 cm), BC (10 cm) equals LM (10 cm), and CA (6 cm) equals MN (6 cm). This gives the correspondence A with N, B with L, and C with M, so triangle ABC is congruent to triangle NLM, making 'A↔N, B↔L, C↔M (ABC≅NLM)' correct. 'A↔L, B↔M, C↔N (ABC≅LMN)' simply matches the vertices in the order they are written without checking the side lengths: AB (8 cm) would need to equal LM (10 cm), which is false. 'A↔M, B↔N, C↔L (ABC≅MNL)' also fails this check, since AB (8 cm) would need to equal MN (6 cm), which is false. 'A↔N, B↔M, C↔L (ABC≅NML)' gets A correct but swaps B and C, so AB (8 cm) would need to equal NM (6 cm), which is also false.
- (d) sin 30°, tan 30°, cos 30° — sin 30° = 1/2 = 0.5, tan 30° = √3/3 ≈ 0.577 and cos 30° = √3/2 ≈ 0.866, so the correct order from smallest to largest is sin 30°, tan 30°, cos 30°. 'sin 30°, cos 30°, tan 30°' swaps the last two, wrongly putting cos 30° before tan 30°. 'cos 30°, tan 30°, sin 30°' is the correct list written backwards, from largest to smallest. 'tan 30°, sin 30°, cos 30°' wrongly swaps sin 30° and tan 30° at the start.
- (d) 50 — The angles in any triangle add up to 180°, so x = 180 − 90 − 40 = 50. "90" comes from subtracting only the 90° angle from 180° and forgetting to also subtract the 40°. "130" comes from adding 90° and 40° together instead of subtracting them from 180°. "45" comes from halving the 90° angle instead of using the angle sum of the triangle at all.
- (c) ∠XYZ — The angle at a named vertex is written with that vertex's letter in the middle, flanked by its two neighbouring vertices. The angle at Y sits between X and Z, its neighbours in quadrilateral WXYZ, so it is written ∠XYZ. ∠WXY names the angle at X, since X is the middle letter, not Y. ∠YZW names the angle at Z, since Z is the middle letter. ∠ZWX names the angle at W, since W is the middle letter.
- (b) 28 — Method: total bricks = (number of squares in the footprint) × (the height of the wall in bricks). Working: the footprint has the row of 5 squares plus the 2 squares in the arm that stands out from the middle of that row, and they do not overlap, giving 5 + 2 = 7 squares; multiplying by the height of 4 bricks gives 7 × 4 = 28. Answer: 28. The distractors: 20 comes from using only the row of 5 and ignoring the arm (5 × 4). 24 comes from treating the bottom square of the arm as if it were one of the row's own squares, so the arm is counted as adding only 1 new square instead of 2 (5 + 1 = 6, then 6 × 4). 32 comes from counting the square of the row directly below the arm a second time as part of the arm (5 + 3 = 8, then 8 × 4).
- (d) £100.48 — Area of the rug = (angle ÷ 360) × π × r² = (180 ÷ 360) × 3.14 × 16 = 0.5 × 50.24 = 25.12 m². Cost = 25.12 × £4 = £100.48. (£200.96 comes from finding the area of a full circle, 3.14 × 16 = 50.24 m², and forgetting the angle fraction before costing it; £25.12 comes from forgetting to square the radius, using 0.5 × 3.14 × 4 = 6.28 m², and then costing that; £50.24 comes from using the arc length formula, 0.5 × 2 × 3.14 × 4 = 12.56, in place of the area, and costing that.)
- (b) (5, −2) — Method: subtract column vectors by subtracting the bottom vector's top number from the top vector's top number, and doing the same for the bottom numbers. Working: top numbers 7 − 2 = 5; bottom numbers 3 − 5 = −2. Answer: u − v = (5, −2). A candidate who drops the negative sign gets (5, 2). A candidate who works out v − u instead of u − v gets (−5, 2). A candidate who adds instead of subtracts gets (9, 8).
- (c) 155° — Turning clockwise adds to the bearing. Starting on a bearing of 065° and turning clockwise through 90° gives 065° + 90° = 155°. A candidate who instead subtracts, working out 90° − 65° = 25°, has performed the wrong operation, giving 025°. A candidate who turns anticlockwise instead of clockwise works out 065° − 90°, which gives a negative number, and adding 360° to fix this gives 335° — the bearing for turning the other way. A candidate who thinks turning does not change the bearing at all keeps the answer as 065°. The new bearing, turning clockwise, is 155°.
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