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.Write down the term for the distance all the way around the edge of a circle.
- 2.Postcard P is similar to postcard Q. The 6 cm side of P corresponds to the 15 cm side of Q, and the 8 cm side of P corresponds to a matching side of Q. Work out the length of that matching side of Q.
- 3.Triangle ABC has a right angle at B and hypotenuse AC = 17 cm, with AB = 8 cm. Triangle DEF has a right angle at E and hypotenuse DF = 17 cm. Which extra fact about DEF would complete the RHS condition for proving that ABC ≅ DEF, with the vertices matching in that order?
- 4.An angle is measured with a protractor as 47°. What type of angle is this?
- 5.Two straight lines AB and CD cross at point O. Angle AOC = 74°. Work out angle AOD.
- 6.A trapezium-shaped allotment plot has two parallel sides that face each other, and two sloping sides that are equal in length (an isosceles trapezium). One of the angles next to the shorter parallel side is 118°. Work out the angle next to the other end of the same shorter parallel side.
- 7.A designer enlarges a drawing of a model car for a poster. She first enlarges the drawing by a scale factor of 1.5, and then enlarges that result by a further scale factor of 2. On the original drawing, the position of a wheel relative to the front bumper is given by the column vector with top number 4 and bottom number −3, in centimetres. What is the corresponding column vector on the poster, in centimetres?
- 8.The diagram shows a prism resting on a table. Which shape is its front elevation, the view looking at it from directly in front?
- 9.A quadrilateral has all four sides equal in length and all four interior angles equal to 90°. Its diagonals are equal in length and bisect each other at right angles. Which quadrilateral matches this description exactly?
- 10.Two triangles are proved congruent using the ASA condition. What can be concluded about the two remaining pairs of corresponding sides that were not part of the original ASA facts?
- 11.Triangles PQS and RQS share the common side QS. PQ equals RQ, and PS equals RS. Which condition proves that triangle PQS is congruent to triangle RQS?
- 12.Write down the name given to a quadrilateral that has exactly one pair of parallel sides.
- 13.A trapezium has parallel sides of 16 cm and 24 cm, and a perpendicular height of 7 cm. Work out the area of the trapezium.
- 14.A polygon has 9 sides. What is the name of this polygon?
- 15.A solid has 6 faces: one pentagon and five triangles that all meet at a single point above the pentagon. What is the name of this solid?
Answer key
- (b) circumference — The distance all the way around the outside edge of a circle is called the circumference. The diameter is a straight line across the circle through the centre, so it is a length through the circle, not around it. The radius is a straight line from the centre to the edge, again a length across, not around. The area is the amount of surface inside the circle, a region, not a length at all.
- (d) 20 cm — The scale factor from P to Q is 15 ÷ 6 = 2.5. Apply the same scale factor to the other side: 8 × 2.5 = 20 cm. A pupil who divides instead of multiplying by the scale factor gets 8 ÷ 2.5 = 3.2 cm. A pupil who adds the difference between the two known sides, 15 − 6 = 9, to 8 instead of scaling gets 8 + 9 = 17 cm. A pupil who rounds the scale factor 2.5 down to 2 gets 8 × 2 = 16 cm. The correct length is 20 cm.
- (d) DE = 8 cm — Method: use the stated correspondence ABC ≅ DEF to work out which side in DEF matches the known side AB in ABC. Working: the correspondence sends A to D, B to E and C to F, so AB corresponds to DE; the right angles at B and E and the equal hypotenuses AC = DF = 17 cm are already given, so DE = 8 cm supplies the third ingredient — Right angle, Hypotenuse, Side. Options: EF = 8 cm matches AB to the wrong side, since EF corresponds to BC, and BC = √(17² − 8²) = 15 cm, not 8 cm; BC = 8 cm states something about triangle ABC rather than the missing fact about DEF, and it is false as well, since BC = 15 cm; angle D = angle A does follow once the triangles are congruent, but RHS is completed by a matching side, not by a matching angle. Answer: DE = 8 cm.
- (b) Acute — An acute angle is any angle less than 90°. Since 47° is less than 90°, it is acute. An obtuse angle is between 90° and 180°, which 47° is not. A reflex angle is greater than 180°, which is far too big for 47°. A right angle is exactly 90°, not 47°. The angle is acute.
- (a) 106° — Angle AOC and angle AOD share the arm OA and together they make up the straight line CD, so they add up to 180°: angle AOD = 180° − 74° = 106°. 74° comes from treating AOC and AOD as vertically opposite angles, which are equal — but they are adjacent angles on a straight line, not vertically opposite. 16° comes from subtracting 74° from 90°, treating the angles as complementary. 148° comes from doubling 74° instead of subtracting it from 180°.
- (d) 118° — In an isosceles trapezium, the two angles next to the same parallel side are equal, because the sloping sides are equal in length. So the angle at the other end of the shorter parallel side also equals 118°.
- (b) (12, −9) — Two enlargements one after the other combine into a single scale factor: 1.5 × 2 = 3. Multiplying a vector by a scalar means multiplying both the top number and the bottom number by it: top = 4 × 3 = 12, bottom = −3 × 3 = −9, giving (12, −9). A candidate who adds the scale factor to each number instead of multiplying gets (4 + 3, −3 + 3) = (7, 0). A candidate who multiplies the top number but leaves the bottom number unchanged gets (12, −3). A candidate who multiplies the bottom number but leaves the top number unchanged gets (4, −9). The correct column vector for the poster is (12, −9).
- (b) Triangle — Method: the front elevation is the outline you see looking straight at the solid's front face. Working: this solid's front face is one of its two identical triangular ends, so looking directly at the front shows exactly that triangular outline. Answer: triangle. The distractors: rectangle comes from picturing the side view instead of the front view — looking along the ridge of the prism, the sloping roof faces project as a rectangle. Trapezium comes from misreading the two sloping roof faces as if together they formed a single four-sided shape. Pentagon comes from tracing round the outline of the whole sketch as it is drawn on the page, which is a five-sided shape, instead of drawing only the view seen looking straight at the front of the solid.
- (b) Square — A square has all four sides equal, all four angles equal to 90°, and diagonals that are equal in length and bisect each other at right angles — every part of the description matches, so Square is correct. A rhombus has all four sides equal and diagonals bisecting at right angles, but its interior angles are not generally 90° (only a square, a special rhombus, has that), so it does not fully match. A rectangle has four 90° angles and equal diagonals, but its sides are not all equal in general, so it fails the equal-sides condition. A kite has two pairs of adjacent equal sides rather than all four sides equal, and its diagonals are not generally equal in length, so it fails both conditions.
- (c) They must also be equal — Once two triangles are proved congruent by any condition, including ASA, they are identical in every respect: every pair of corresponding sides and every pair of corresponding angles must be equal, not just the ones originally used to prove the congruence. So the two remaining pairs of corresponding sides must also be equal, making 'they must also be equal' correct. 'They might be equal or not' and 'not enough information to say' both wrongly suggest that congruence only guarantees the specific facts used to prove it, when congruence actually guarantees the triangles are identical overall. 'They must be different' is backwards: the triangles being identical is the entire point of proving congruence, not a reason for a side to differ.
- (d) SSS, using shared side QS — PQ equals RQ and PS equals RS are two given pairs of equal sides, and QS is common to both triangles, so QS equals itself and gives a third pair of equal sides. Three pairs of equal sides is exactly the SSS condition, so 'SSS, using shared side QS' is correct. 'SAS, using the angle at Q' is wrong because no angle is given anywhere in this question; angle PQS and angle RQS are not stated to be equal, and assuming they are would be assuming the very thing being proved. 'Only two pairs of sides — not enough' is wrong because it forgets that the shared side QS is itself a third pair of equal sides. 'Cannot prove — no angle given' is wrong because SSS is one of the four basic congruence conditions and specifically requires no angle at all.
- (c) Trapezium — A trapezium is defined as a quadrilateral with exactly one pair of parallel sides.
- (b) 140 cm² — Method: the area of a trapezium is the mean of the two parallel sides multiplied by the perpendicular height. Working: (16 + 24) ÷ 2 = 20, then 20 × 7 = 140. Answer: 140 cm². The distractors: 280 cm² comes from multiplying the sum of the parallel sides by the height, (16 + 24) × 7, and forgetting to halve; 168 cm² comes from using only the longer parallel side, 24 × 7, as though the shape were a rectangle; 47 cm comes from adding all three given lengths, 16 + 24 + 7, which gives a length rather than an area.
- (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.
- (c) Pentagonal pyramid — Method: a pyramid has one base and triangular faces that all meet at a single apex; the base shape gives the pyramid its name. Working: the base is a pentagon and the other five faces are triangles meeting at one point, so this is a pyramid with a pentagon base. A student who answers pentagonal prism has confused a pyramid, whose sloping faces meet at an apex, with a prism, which has two identical parallel faces. A student who answers hexagonal pyramid has miscounted the base as having 6 sides instead of 5. A student who answers triangular pyramid has misread the five triangular side faces as meaning the base itself is a triangle. Answer: pentagonal pyramid.
Build your own mix at the worksheet builder.