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.
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Geometry and measures worksheet — GCSE Foundation
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- 1.Each of these has an exact value. Write down the one whose value is the greatest.
- 2.p is the column vector with top number 4 and bottom number −3. Work out 2p, giving your answer as a column vector in the form (top, bottom).
- 3.Triangle ABC has AB = (2x + 3) cm, BC = 11 cm and angle B = 90°. Triangle DEF has DE = 13 cm, EF = 11 cm and angle E = 90°. Given that the two triangles are congruent by SAS, work out the value of x.
- 4.Two sides of a triangle are 9 cm and 15 cm long. Write down which one of these lengths is possible for the third side.
- 5.A cake recipe takes 135 minutes from start to finish. Write this time as hours and minutes.
- 6.In parallelogram ABCD the vertices are labelled in order round the shape, so angle A and angle B are at the two ends of the same side. Angle A is 70°. Work out the size of angle B.
- 7.Lines AB and CD cross at a point, forming four right angles where they meet. Which notation correctly describes the relationship between AB and CD?
- 8.A ramp rises at 30° to the horizontal. Its sloping surface is 4.8 m long. Safety rules say the vertical rise of a ramp must be no more than 2.5 m. Work out how far below that limit the rise of this ramp is.
- 9.Triangle DEF is isosceles, with DE = DF. Angle E = 58°. Give a reason why angle F = 58°.
- 10.Triangle PQR has vertices P(0, 0), Q(9, 0) and R(0, 4). Work out the area of triangle PQR.
- 11.A map has a scale of 1 : 25000. A distance between two points measures 2 cm on the map. What is the real distance, in kilometres?
- 12.A children's slide is built as a right-angled triangle: the vertical ladder side is 2.4 m, the horizontal base along the ground is 3.2 m, and the sloping slide is the third side. Work out the exact length of the sloping slide.
- 13.A quadrilateral has vertices A(1, 1), B(5, 1), C(6, 4) and D(2, 4). By comparing the y-coordinates of A, B and of D, C, write down whether the sides AB and DC are parallel, and give a reason for your answer.
- 14.A sector of a circle is bounded by two straight edges and one curved edge. Write down the mathematical term for each of the two straight edges.
- 15.Write down the column vector that describes a translation of 5 units to the left and 3 units down.
Answer key
- (c) tan 45° — Method: replace each ratio by its exact value, then compare. Working: a right-angled triangle with a 45° angle is isosceles, so its opposite and adjacent sides are equal and the tangent of 45° is exactly 1. The others are cos 30° = √3/2, about 0.87; sin 45° = √2/2, about 0.71; and cos 60° = 1/2. Answer: tan 45°, the only one of the four that reaches 1. Reading √3/2 as though it were √3, about 1.73, makes cos 30° look the largest, but the division by 2 is part of the value. Ranking by the size of the angle also fails here, because the cosine of an angle falls as the angle grows.
- (b) (8, −6) — Multiplying a column vector by a scalar multiplies both numbers in the vector by that scalar: 2p = (2×4, 2×(−3)) = (8, −6). (8, −3) doubles the top number but leaves the bottom number, −3, unchanged. (4, −6) doubles the bottom number but leaves the top number, 4, unchanged. (8, 6) doubles both numbers correctly in size but loses the negative sign on the bottom number.
- (d) 5 — Method: since the triangles are congruent by SAS, the corresponding sides AB and DE must be equal, because both are the side next to the given right angle that is not BC or EF. Working: AB = DE gives 2x + 3 = 13, so 2x = 10, so x = 5. Options: 10 comes from dropping the coefficient of x and solving x + 3 = 13 instead of 2x + 3 = 13; 4 comes from matching AB to the wrong side, EF, giving 2x + 3 = 11, so 2x = 8, so x = 4; 8 comes from a sign error, solving 2x − 3 = 13 instead of 2x + 3 = 13, giving 2x = 16, so x = 8. Answer: 5.
- (d) 10 cm — For a triangle to exist, any two sides must add up to more than the third side. 9 + 10 = 19 > 15, and 15 − 9 = 6 < 10, so 10 cm satisfies the triangle inequality. The other lengths fail: 6 cm gives 9 + 6 = 15, which is not more than 15; 24 cm and 26 cm are each at least as large as 9 + 15 = 24.
- (c) 2 hours 15 minutes — Divide the total minutes by 60: 135 ÷ 60 = 2 remainder 15, so that is 2 hours 15 minutes. Treating 100 minutes as one hour, a metric-style mistake, gives 135 − 100 = 35, so 1 hour 35 minutes. Subtracting 60 twice to reach the 15 minutes left over, but losing count and recording only one of the two hours removed, gives 1 hour 15 minutes. Writing 135 ÷ 60 = 2.25 and then reading the '25' as minutes, instead of converting the 0.25 of an hour into 15 minutes, gives 2 hours 25 minutes.
- (a) 110° — Method: in a parallelogram the two angles at the ends of one side are co-interior angles between a pair of parallel sides, so they add up to 180°. Working: angle A + angle B = 180°, so angle B = 180° − 70° = 110°. Answer: 110°. The distractors: 70° comes from applying the rule for opposite angles of a parallelogram, which are equal, to two angles that are next to each other instead; 20° comes from treating the two angles as complementary and working out 90° − 70°; 290° comes from using the 360° angle sum of a quadrilateral and taking away only the one angle that is given, 360° − 70°.
- (c) AB ⊥ CD — Method: match the description of the lines to the correct symbol. Working: the symbol ⊥ means 'is perpendicular to', used when two lines meet at right angles, which is exactly what is described. Options: ∥ means 'is parallel to', for lines that never meet, so it does not apply here; = compares two lengths as equal, but no lengths are mentioned; ≅ means 'is congruent to', used for whole shapes, not for describing how two lines cross. Answer: AB ⊥ CD.
- (a) 0.1 m — Method: the sloping surface is the hypotenuse and the vertical rise is the side opposite the 30° angle, so rise = 4.8 × sin 30°; then compare that rise with the limit. Working: the exact value of sin 30° is one half, so the rise = 4.8 × 1/2 = 2.4 m. The limit is 2.5 m, and 2.5 − 2.4 = 0.1. Answer: the ramp is 0.1 m below the limit. Working out the rise and stopping there gives 2.4 m, which answers a question that was not asked. Dividing by sin 30° instead of multiplying gives 4.8 ÷ 0.5 = 9.6 and then 9.6 − 2.5 = 7.1 m. Treating sine as proportional to the angle, so that sin 30° is a third of sin 90°, gives 4.8 ÷ 3 = 1.6 and then 2.5 − 1.6 = 0.9 m.
- (d) Base angles of an isosceles triangle are equal — DE = DF, so triangle DEF is isosceles with DE and DF as the two equal sides. The base angles opposite those equal sides, angle E and angle F, are therefore equal to each other. Angle F = 58°. A student who instead quotes 'Angles in a triangle sum to 180°' has picked a true fact about triangles, but that fact finds a missing angle from the other two — it does not explain why two angles are equal to each other. A student who quotes 'Angles on a straight line sum to 180°' has confused this with a straight-line angle fact, but no straight line of angles is described in this triangle.
- (a) 18 — PQ lies along the x-axis with length 9, and PR lies along the y-axis with length 4, and these two sides meet at right angles at P, so they can be used as the base and height of the triangle. Area = 1/2 × base × height = 1/2 × 9 × 4 = 18. 36 comes from multiplying the base and height but forgetting to halve the result. 13 comes from adding the two lengths, 9 + 4, instead of multiplying them. 26 comes from the perimeter-style calculation 2 × (9 + 4) instead of the triangle area formula.
- (c) 0.5 — The real distance is 2 × 25000 = 50000 cm. Converting to metres, by dividing by 100, gives 500 m, and converting to kilometres, by dividing by 1000, gives 0.5 km. A candidate who divides the 50000 cm by 1000 in one go, applying the metres-to-kilometres factor straight to the centimetres, gets 50 km. A candidate who divides by 100 twice, treating 100 m as 1 km, gets 5 km. A candidate who slips one extra decimal place when converting 500 m to kilometres gets 0.05 km. The real distance is 0.5 km.
- (d) 4 m — The sloping slide is the hypotenuse of a right-angled triangle with the other two sides 2.4 m and 3.2 m. By Pythagoras' Theorem, hypotenuse² = 2.4² + 3.2² = 5.76 + 10.24 = 16. Square root: √16 = 4 m. Adding the two sides instead of squaring them gives 2.4 + 3.2 = 5.6 m. Truncating each squared side to a whole number before adding (5.76 to 5 and 10.24 to 10) gives √15 ≈ 3.87 m. Forgetting to take the square root and leaving the sum of the squares as the answer gives 5.76 + 10.24 = 16, written as 16 m.
- (b) Parallel, since AB and DC are both horizontal lines — A and B both have y-coordinate 1, so AB is horizontal; D and C both have y-coordinate 4, so DC is horizontal too. Two horizontal lines are always parallel, so AB and DC are parallel. "Not parallel, since AB and DC have different lengths" is wrong twice over: AB runs from x = 1 to x = 5 and DC from x = 2 to x = 6, so both are in fact 4 units long, and in any case length has no bearing on whether two lines are parallel — only direction does. "Not parallel, since AB is horizontal and DC is vertical" misreads the coordinates of D and C, which share the y-coordinate 4 and so give a horizontal line, not a vertical one. "Parallel, since A and D have the same x-coordinate" rests on a false claim: A has x-coordinate 1 and D has x-coordinate 2, so they do not share an x-coordinate — and even if they did, it would say nothing about whether AB is parallel to DC.
- (c) Radius — Method: recall that a sector is formed using two straight lines drawn from the centre out to the circle's edge. Working: each straight edge of a sector runs from the centre of the circle to a point on the circumference. A student who answers chord has confused a straight edge from the centre with one joining two points on the circumference. A student who answers diameter has wrongly assumed the two straight edges must form a single full diameter. A student who answers arc has named the curved edge instead of the straight edges. Answer: radius (radii).
- (d) $\binom{−5}{−3}$ — Left is a negative horizontal move and down is a negative vertical move, so the vector is $\binom{−5}{−3}$. $\binom{5}{−3}$ comes from treating 'left' as a positive move. $\binom{−5}{3}$ comes from treating 'down' as a positive move. $\binom{−3}{−5}$ comes from swapping the horizontal and vertical components.
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