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
MathsUKwww.geekhero.co.uk
- 1.In the RHS congruence condition, which three facts must be true about two triangles for RHS to prove they are congruent?
- 2.Triangles LMN and XYZ have LM = XY, MN = YZ and LN = XZ. Give a reason why triangle LMN is congruent to triangle XYZ.
- 3.Three angles meet at a single point on one side of a straight line. Two of the angles are 48° and 65°. Work out the third angle.
- 4.A hiker walks due west. What is the three-figure bearing for this direction?
- 5.A chord divides a circle into two regions, one larger than the other. Write down the term used for the smaller of these two regions.
- 6.A polygon has 9 sides. What is the name of this polygon?
- 7.A flower bed in the shape of a parallelogram has a base of 3.5 m and a perpendicular height of 2 m. Compost costs £4.80 per m². Work out the total cost of covering the flower bed with compost.
- 8.A rhombus has all four sides equal in length. Write down how many pairs of parallel sides a rhombus has.
- 9.A regular nonagon has 9 sides. Work out the sum of its interior angles.
- 10.A cuboid measures 5 cm long (left to right), 3 cm deep (front to back) and 2 cm tall. Its front elevation is 5 cm wide by 2 cm high. Its side elevation is 3 cm wide by 2 cm high. What are the dimensions of its plan view, looking down from above?
- 11.The curved part of a circle's circumference, between two points on the circle, has its own name. Write down this name.
- 12.A hexagon has interior angles of 130°, 142°, 125°, 150°, 135° and x°. Work out the value of x.
- 13.A triangular prism has two triangular end faces and three rectangular faces joining them. How many faces does it have in total?
- 14.In a right-angled triangle, θ is one of the acute angles. Write down the trigonometric ratio, in terms of the opposite and adjacent sides, that is used to find θ.
- 15.A photograph is 10 cm wide and 15 cm tall. It is enlarged to make a similar poster that is 40 cm wide. The poster costs £0.80 per centimetre of its height to print, based on its full height. Work out the cost of printing the poster.
Answer key
- (b) Right angle, equal hypotenuse, one more side — RHS requires a right angle in each triangle, equal hypotenuses, and one further pair of equal sides, so 'right angle, equal hypotenuse, one more side' is correct. 'Right angle, hypotenuse, one more angle' is wrong because it asks for an extra equal angle instead of an extra equal side, which RHS does not check. 'Two sides and the angle between them' describes SAS, not RHS, so it is wrong. 'Right angle and all three sides equal' describes SSS with an extra right-angle condition, asking for more information than RHS actually needs, so it is wrong.
- (a) SSS – all three corresponding sides are equal — All three pairs of corresponding sides are stated as equal — LM = XY, MN = YZ and LN = XZ — with no angle mentioned. This matches the SSS condition, so triangle LMN is congruent to triangle XYZ.
- (c) 67° — Angles on a straight line add up to 180°. Add the two known angles: 48° + 65° = 113°. Subtract from 180°: 180° − 113° = 67°.
- (b) 270° — Bearings are measured clockwise from north (000°). Due north is 000°, due east is 090°, due south is 180° and due west is 270°. Choosing 090° gives the bearing for due east, not west. Choosing 180° gives the bearing for due south. Choosing 000° gives the bearing for due north.
- (c) minor segment — A chord splits a circle into two segments; the smaller of the two is called the minor segment and the larger one the major segment. "major segment" names the larger region, the opposite of what is asked for. "sector" is a different region altogether, enclosed by two radii and an arc, not by a chord. "arc" is a curved length along the circumference, not an enclosed region at all.
- (b) Nonagon — Method: match the number of sides given to the correct polygon name. Working: a polygon with 9 sides is called a nonagon. An octagon has 8 sides, a decagon has 10 sides, and a heptagon has 7 sides, so each names a different number of sides from the 9 given. Answer: nonagon.
- (b) £33.60 — The area of the parallelogram flower bed is base × height = 3.5 × 2 = 7 m². The cost is 7 × £4.80 = £33.60. £16.80 comes from using the triangle formula instead of the parallelogram formula: 3.5 × 2 = 7, and half of 7 is 3.5 m², then 3.5 × £4.80 = £16.80. £26.40 comes from adding the base and height, 3.5+2 = 5.5, instead of multiplying them, then multiplying by £4.80. £7.00 correctly finds the area, 7 m², but forgets to multiply it by the cost per m².
- (b) 2 — A rhombus is a special parallelogram, so opposite sides are parallel — that is two pairs of parallel sides. A quadrilateral with only one pair of parallel sides is a trapezium, not a rhombus, so 1 is wrong. A kite has 0 pairs of parallel sides, not a rhombus. There are only two pairs of opposite sides in a quadrilateral altogether, so 4 is not possible. The correct answer is 2.
- (a) 1260° — The sum of the interior angles of a polygon with n sides is (n − 2) × 180°. For a nonagon, n = 9, so the sum is (9 − 2) × 180° = 7 × 180° = 1260°. 1620° uses 9 × 180° without subtracting 2 from n first. 140° is the size of a single interior angle of a regular nonagon (1260° ÷ 9), not the sum of all nine. 1440° uses (n − 1) × 180° = 8 × 180° instead of (n − 2) × 180°.
- (b) 5 cm by 3 cm — The plan view looks straight down on the cuboid's footprint, so it shows the length (5 cm, left to right) and the depth (3 cm, front to back) — the two dimensions that do not involve height. "5 cm by 2 cm" repeats the front elevation's dimensions, pairing the length with the height instead of the depth. "3 cm by 2 cm" repeats the side elevation's dimensions, again pairing the depth with the height rather than with the length. "5 cm by 5 cm" comes from mistakenly assuming the plan must be a square, pairing the length with itself instead of with the depth.
- (a) arc — A curved part of a circle's circumference, between two points, is called an arc. A chord is the straight line joining those two points, not the curved part. A segment is the region enclosed between a chord and an arc, and a sector is the region enclosed between two radii and an arc — both segment and sector are regions, not curved lengths.
- (c) 38° — The interior angles of a hexagon sum to (6 − 2) × 180° = 720°. 130° + 142° + 125° + 150° + 135° = 682°. x = 720° − 682° = 38°. A student who uses 6 × 180° instead of (6 − 2) × 180°, forgetting to subtract the 2, gets 1080° − 682° = 398°. A student who mis-adds the five given angles as 680° instead of 682° gets x = 720° − 680° = 40°. A student who answers with one of the given angles instead of solving for x might write 142°.
- (a) 5 — Method: add the two triangular end faces to the three rectangular side faces. Working: 2 triangular faces + 3 rectangular faces = 5 faces in total. A student who answers 6 has confused this with a cuboid's face count. A student who answers 9 has counted the edges instead of the faces. A student who answers 3 has only counted the rectangular faces and forgotten the two triangular ends. Answer: 5 faces.
- (c) tan θ = opposite/adjacent — The tangent ratio is defined as tan θ = opposite/adjacent, so this is the ratio that connects the opposite and adjacent sides. "sin θ = opposite/adjacent" wrongly labels this ratio as sine, when sine actually connects the opposite side and the hypotenuse. "cos θ = opposite/adjacent" wrongly labels it as cosine, when cosine connects the adjacent side and the hypotenuse. "tan θ = adjacent/opposite" uses the correct ratio name but has the opposite and adjacent sides the wrong way round.
- (d) £48 — Scale factor = new width ÷ original width = 40 ÷ 10 = 4. Poster height = 15 × 4 = 60 cm. Cost = 60 × £0.80 = £48. (£12 comes from forgetting to scale the height at all, and pricing the original 15 cm height; £15.20 comes from adding the scale factor 4 to the height instead of multiplying by it; £3 comes from dividing the height by the scale factor instead of multiplying by it.)
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