Worksheet: Geometry and measures (easy)

Answer key

1. (b) 0 — At 0°, the opposite side of the right-angled triangle has shrunk to zero length, so sin 0° = 0. 1 is the value of sin 90° (mixing up the two angles). 1/2 is the value of sin 30°. 'Undefined' is what happens for tan 90°, not sin 0° — sin 0° has a perfectly good exact value.
2. (a) a circle of radius 4 cm, centred at O — Every point exactly 4 cm from a fixed point O sweeps out a full circle of radius 4 cm centred at O — that is the definition of this locus. "a straight line 4 cm long, starting at O" only covers points in one direction, not every direction. "a square with sides of 4 cm, centred at O" would only touch the true locus at a few points on its perimeter, since most of a square's edge is not 4 cm from its centre. "two points, each 4 cm from O" comes from picturing only the points directly left and right of O, forgetting every other direction round it.
3. (a) Infinitely many — Method: a radius is any straight line from the centre of a circle to a point on the circle, so counting the radii means counting the possible end points, and those end points are the points of the circle itself. Working: a circle is a continuous curve, so between any two points on it there is always another point; the supply of end points therefore never runs out, and each end point gives one radius. Every one of those radii is the same length, because equal distance from the centre is what makes the curve a circle in the first place. Answer: Infinitely many. The distractors: Exactly one comes from treating a radius as a single fixed line, because a textbook diagram usually shows only one drawn in; Exactly two comes from picturing a diameter and counting the two halves it is made of; Exactly four comes from picturing the circle cut into quarters by two perpendicular diameters and counting the four radii that appear in that picture.
4. (b) bisect the angle between the two lines — The points equidistant from two straight lines that meet lie on the angle bisector of the angle between them — every point on a bisector is the same perpendicular distance from each line, which is exactly what the angle bisector construction produces. "construct the perpendicular bisector of the two lines" confuses the bisector of a line SEGMENT between two points with the bisector of an ANGLE between two lines — a different construction for a different kind of equidistance. "construct a perpendicular from the crossing point" only gives one new line at 90° to one of the originals, not the points equidistant from both. "draw a circle centred at the crossing point" gives points a fixed distance from the crossing point, not points equidistant from the two lines.
5. (c) (2, 4) and (6, 4) — Method: two points lie on the same horizontal line when they stand at the same height above the x-axis, and a point's height is given by its y-coordinate, the second number in the bracket. Working: compare the second numbers in each pair. For (3, 5) and (3, 8) they are 5 and 8, which differ. For (1, 2) and (2, 1) they are 2 and 1, which differ. For (0, 0) and (1, 1) they are 0 and 1, which differ. For (2, 4) and (6, 4) they are 4 and 4, which agree, so only that pair lies on a horizontal line. Answer: (2, 4) and (6, 4). The distractors: (3, 5) and (3, 8) comes from swapping horizontal for vertical, since equal x-coordinates place two points one above the other; (1, 2) and (2, 1) comes from looking for the same pair of numbers rather than for the same y-coordinate; (0, 0) and (1, 1) comes from reading 'horizontal' as simply 'in a straight line', which any two points are.
6. (b) √2/2 — cos 45° comes from a right-angled isosceles triangle with both shorter sides 1 and hypotenuse √2, giving cos 45° = 1/√2 = √2/2. 1/2 is the value of cos 60° (mixing up the two angles). √2 is the hypotenuse length itself, not divided by it (forgetting to divide by the hypotenuse). √3/2 is the value of cos 30° (using the wrong special triangle).
7. (b) SAS — Two pairs of matching sides are equal (AB = DE and BC = EF) and the angle between them is equal (angle ABC = angle DEF), which is exactly the Side-Angle-Side congruence test: SAS. SSS is wrong because only two pairs of sides are given, not three. ASA is wrong because only one angle is given, and it sits between the two sides rather than the side sitting between two angles. RHS is wrong because no right angle is mentioned in either triangle.
8. (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.
9. (a) 143° — Angles on a straight line add up to 180°. Set up 37° + x = 180°. Subtract: x = 180° − 37° = 143°. 53° comes from using 90° as the total, as if the two angles made a right angle, instead of the 180° of a straight line.
10. (c) (−2, 10) — To translate M(3, 8) by $\binom{−5}{2}$, add −5 to the x-coordinate and 2 to the y-coordinate: (3 + (−5), 8 + 2) = (−2, 10). (5, 3) comes from swapping the two components of the vector before applying them. (−2, 8) applies only the x-component and forgets to add the y-component. (3, 10) applies only the y-component and forgets to add the x-component.
11. (a) 0 — tan 0° = 0, because there is no opposite side to consider when the angle itself is 0° — one of the exact values you need to know. 1 is the exact value of tan 45°, not tan 0°. √3 is the exact value of tan 60°. 1/√3 is the exact value of tan 30°.
12. (d) 4 — Method: use the definition of a rectangle — a quadrilateral in which every interior angle is a right angle. Working: a quadrilateral has four interior angles, and in a rectangle all four of them measure 90°, which is consistent with the 360° angle sum because 4 × 90° = 360°. Answer: 4. The distractors: 2 comes from applying the rule that opposite angles are equal and concluding that only one pair of corners is square; 1 comes from thinking that a single square corner is enough to make a shape a rectangle; 0 comes from confusing a rectangle with a general parallelogram, which has no right angles unless it is a rectangle.
13. (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.
14. (c) Scalene — Method: equal angles in a triangle sit opposite equal sides, so compare the three angles with each other. Working: 10°, 58° and 112° are all different, so no two sides of the triangle are equal either, and a triangle with no equal sides is scalene. Answer: scalene. The distractors: isosceles is chosen by candidates who pair up the two acute angles, 10° and 58°, as base angles without checking that they are actually equal; equilateral is chosen by candidates who check that the three angles add to 180° and take that as meaning the triangle is regular; right-angled is chosen by candidates who see that 112° is larger than 90° and classify the triangle as containing a right angle, when in fact none of the three angles is 90°.
15. (b) 2 cm — Method: a diameter is made of two radii end to end, so going back from a diameter to a radius undoes that doubling, which gives radius = diameter ÷ 2. Working: the diameter is 4 cm, so the radius is 4 ÷ 2 = 2 cm. Answer: 2 cm. The distractors: 8 cm comes from multiplying by 2 instead of dividing by it, the relationship applied in the wrong direction; 1 cm comes from halving twice, once to reach the radius and then once more as though a second halving were called for; 0.5 cm comes from writing the division upside down as 2 ÷ 4 rather than 4 ÷ 2.