Printable · GCSE Higher · ages 14-16
Geometry and measures worksheet — GCSE Higher
Fifteen questions across the geometry and measures statements at Higher tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
Non-calculator
Answer key: Geometry and measures worksheet — GCSE Higher
MathsUKwww.geekhero.co.uk
- (c) 58° — In an isosceles triangle, the base angles opposite the equal sides are equal. Since DE = DF, angle F is the base angle equal to angle E, so angle F = 58°.
- (c) 1/4 — A full turn at the centre of a circle is 360°, so a sector's fraction of the circle is its angle divided by 360°: 90 ÷ 360 = 1/4. 1/2 would be the fraction for a sector with an angle of 180°, not 90°. 3/4 is the fraction of the rest of the circle, the major sector left over from the 270° that is not part of this sector. 1/8 would be the fraction for a sector with an angle of 45°, half of 90°.
- (d) A rectangle — Method: to find the plan view, work out the outline traced when looking straight down onto the solid from directly above, not the outline shown in the angled sketch. Working: this solid has two identical flat round ends joined by one curved surface, and it is lying on its side rather than standing upright; viewed from above, the curved surface gives two straight edges running the full length of the solid, the width of the round ends apart, and each flat round end — seen edge-on from directly above — also becomes a straight edge of that same width, with no curve remaining. Four straight edges, with opposite sides equal and meeting at right angles, form a rectangle. Answer: a rectangle. The distractors: a circle comes from picturing the solid as if it were standing upright on one of its flat ends, giving the plan of an upright version instead of working out the plan of the solid as it actually lies. An oval comes from copying the foreshortened shape of a round end as it is drawn in the angled sketch, instead of working out the true shape seen from directly above, which has no such foreshortening. A rectangle with rounded ends comes from carrying the curve of the round ends over into the plan view, when in fact a flat round end viewed edge-on from directly above shows no curve at all, only a straight edge.
- (d) 2 — Method: if f is 3 times e, then each part of f equals 3 times the matching part of e. Working: using the bottom numbers, 6 = 3 × k, so k = 2. Answer: k = 2. A candidate who multiplies instead of dividing, working out 6 × 3, gets 18. A candidate who uses the top numbers' ratio instead, 15 ÷ 5, and gives that ratio as k gets 3. A candidate who adds instead of using the multiple relationship, working out 6 + 3, gets 9.
- (a) 118° — When two straight lines cross, the angles that are vertically opposite each other are always equal. So the angle vertically opposite 118° is also 118°. A candidate who instead finds the angle next to it on the straight line, using 180° − 118° = 62°, has found the adjacent angle, not the vertically opposite one. A candidate who answers 180° has confused the rule with angles on a straight line. A candidate who doubles the angle, giving 236°, has applied no valid angle rule at all. The vertically opposite angle is 118°.
- (a) the lens-shaped region where the two circles overlap — The region within 30 km of the ship is a circle of radius 30 km around the ship; the region within 20 km of the lighthouse is a circle of radius 20 km around the lighthouse. Since 30 + 20 = 50 km is more than the 40 km between them, and 30 − 20 = 10 km is less than 40 km, the two circles genuinely overlap, in a lens-shaped region where both conditions hold at once. "the whole smaller circle, since it fits inside the larger one" would only be true if the 40 km distance between the centres were small enough for one circle to swallow the other, which it is not here. "no overlap — 30 km and 20 km don't add to 40 km" wrongly assumes the two radii must add up to exactly the distance between the centres for an overlap to exist; any total greater than that distance is enough. "a straight strip, exactly halfway between the two" confuses this with an equidistant locus, which is a different problem from the one asked here.
- (b) angle B = angle E — AB and BC meet at vertex B, so the angle INCLUDED between them is angle B; making angle B = angle E completes SAS. Angle A sits between AB and AC, not between AB and BC, so it is not the included angle needed for SAS here. Angle C sits between BC and CA, not between AB and BC, so it is not included either. AC = DF would give a third pair of equal sides, which proves congruence by SSS instead of SAS.
- (a) 310° — Method: give the two rotations opposite signs since they turn in opposite senses — clockwise positive, anticlockwise negative — combine them into a single signed turn, then convert that turn into an angle measured clockwise between 0° and 360°. Working: the first rotation is 200° clockwise, so +200. The second is 250° anticlockwise, so −250. Combined: 200 − 250 = −50, meaning the net effect is a 50° turn anticlockwise. Measured clockwise instead, that same turn is 360 − 50 = 310°. Answer: 310°. Give the two rotations opposite signs before combining them, and convert a negative (anticlockwise) result into a clockwise angle by subtracting it from 360°, not from 180°: taking 50° away from a half turn gives 130°, which is a different rotation altogether; adding the two sizes as if both were clockwise gives 450°, which reduces to 90°; and reporting the anticlockwise size without converting it gives 50°.
- (a) (4, −4) — A point is invariant under a reflection only if it lies exactly on the mirror line. The line y = −x consists of every point where the y-coordinate is the negative of the x-coordinate: (4, −4) satisfies this, since −4 = −(4), so it is invariant. (5, 5) lies on the line y = x, a different line altogether, not y = −x. (4, 4) has equal coordinates, but that alone does not put it on y = −x; it would need y = −4, not 4. (−4, −4) also has equal coordinates and lies on y = x, not y = −x — its coordinates would need opposite signs to sit on the given mirror line. Only a point whose coordinates are negatives of each other stays fixed under this reflection.
- (a) 110° — The truss is isosceles, so its two base angles are equal; since the three angles of the triangle add up to 180° and the apex is 40°, each base angle is (180 − 40) ÷ 2 = 70°. The angle between the sloping edge and the joist, extended beyond the base of the truss, sits on a straight line with that 70° base angle, and angles on a straight line add up to 180°, so the required angle is 180 − 70 = 110°. "70°" comes from stopping after finding the base angle and giving it directly, without also using the straight-line fact to find the angle on the OUTSIDE of the truss. "140°" comes from doubling the base angle instead of using the straight-line fact. "20°" comes from taking half of the apex angle (40° ÷ 2), mistaking it for the required angle instead of properly using the triangle's angle sum.
- (a) 1/3 — Method: since OABC is a parallelogram, B = OA + OC = a + c. M = (1/2)c, since M is the midpoint of OC. Since AN is twice NC, N is 2/3 of the way along AC from A, so N = a + 2/3(c − a) = (1/3)a + (2/3)c. Working: MN = N − M = (1/3)a + (1/6)c, and MB = B − M = a + (1/2)c. Comparing term by term, 1/3 × (a + (1/2)c) = (1/3)a + (1/6)c, which matches MN exactly. Answer: k = 1/3, so M, N and B lie on a straight line. Giving 2/3 instead is the scalar linking N to B (NB = (2/3)MB), not M to N; giving 1/6 is just MN's c-coefficient read off on its own, without comparing it to MB's c-coefficient at all; and giving 3 is the scalar the wrong way up — it is MB that equals 3 × MN, not the other way round, since MN = k × MB was what was asked for. Always match the direction of the scalar to the vectors exactly as the question states them.
- (a) 9 — A cube has 9 planes of symmetry in total: 3 that pass through the middles of pairs of opposite faces, and 6 more that pass through pairs of opposite edges diagonally. A candidate who counts only the 3 face-to-face planes — which is correct for a cuboid with three different edge lengths, but forgets that a cube's equal edges create 6 more diagonal planes — answers 3. A candidate who counts only the 6 diagonal planes and forgets the 3 face-to-face ones answers 6. A candidate who confuses the number of planes of symmetry with the number of edges on a cube answers 12. The correct total for a cube is 9.
- (c) 40.5 kg — Since the statues are made from the same material, mass is proportional to volume, which scales with the cube of the length scale factor. The length scale factor is 45 ÷ 30 = 1.5, so the volume (and mass) scale factor is 1.5³ = 3.375. The larger statue's mass is 12 × 3.375 = 40.5 kg. 18 kg comes from multiplying by the length factor 1.5 directly. 27 kg comes from using the area scale factor 1.5² = 2.25 instead of the volume scale factor. 54 kg comes from treating 'cubed' as 'multiplied by 3', giving 1.5 × 3 = 4.5 instead of 1.5³.
- (c) 115 — Method: use corresponding angles to carry the 65° angle from the lower rafter up to the upper rafter, then use angles on a straight line to move to the other side of the strut. Working: the angle above the upper rafter and to the left of the strut corresponds to the given angle, so it is 65°; the angle above the upper rafter and to the right of the strut lies on a straight line with it, so it is 180 − 65 = 115. Answer: 115°. A candidate who assumes the angle stays 65° without allowing for the move from the left of the strut to the right of it gives 65. A candidate who uses 90° instead of 180°, working out 90 − 65, gets 25. A candidate who adds instead of subtracting, working out 180 + 65, gets 245.
- (b) 21 cm — By Pythagoras' theorem, the other side = √(29² − 20²) = √(841 − 400) = √441 = 21 cm. "9 cm" comes from subtracting the two given lengths directly, 29 − 20 = 9, instead of subtracting their squares. "441 cm" is the value under the square root sign, correct as far as it goes but with the final square root step left out. "35 cm" comes from adding the squares of the two given lengths instead of subtracting them, √(29² + 20²) = √1241 ≈ 35, treating both given lengths as if they were the two shorter sides rather than a shorter side and the hypotenuse.
Build your own mix at the worksheet builder.