Reasoning Questions

Congruence and similarity 1 Show that triangle ABC and triangle ADE are similar. [3] Angle ABC = angle ADE (corr. ∠, BC//DE) [1] Angle BAC = angle DAE (common ∠) [1] ∴ triangle ABC and triangle ADE are similar. (AA Test) (shown) [1] 2 Show that triangle ABC and triangle DEC are similar. [3] Angle ABC = angle DEC (angle in same segment) [1] Angle ACB = angle DCE (vert. opp. ∠) [1] ∴ triangle ABC and triangle DEC are similar. (AA Test) (shown) [1]

3 Show that triangle AOC and triangle BOC are congruent. [4] Angle OAC = angle OBC (radius ⊥ tangent) (R) [1] OC (common hypotenuse) (H) [1] OA = OB (radius) (S) [1] ∴ triangle AOC and triangle BOC are congruent. (RHS Test) (shown) [1]

Pythagoras Theorem/Right-angled triangle 1 Show that angle ABC = 90°. [3] AB2 + BC2 = 82 + 152 = 289 [1] AC2 = 172 = 289 [1] Since AB2 + BC2 = AC2, by converse of Pythagoras Theorem, hence triangle ABC is a right-angle triangle and angle ABC = 90° (shown) [1] 2 (O level 2023 P1 Qn 14) ABC is a triangle. D is a point on AC such that it is equidistant from A, B and C. Explain why angle ABC is a right angle. (Ans from theannexproject)

Data Analysis 1 (O level 2023 P1 Qn 16) (Ans from theannexproject)

2 The bar graph shows the number of books read by two class, A and B this week. State one aspect of the graph that may be misleading and explain how this may lead to a misinterpretation of the graph. [2] The y-axis does not start from 0. Based on the graph, the number of books read by Class B is thrice the number of books read by Class A. However, it should be twice.

Numbers/Prime factorisation 1 The prime factorisation of two integers, M and N are shown below. M = 𝑝8 × 𝑞4 × 𝑟6 N = 𝑝𝑥+1 × 𝑞4 × 𝑟2𝑦−1 Explain why M is a perfect square? [1] The powers of p, q and r are multiples of 2. 2 Show that (2𝑛 + 3)2 − 5 is a multiple of 4 for all integer values of n. [2] (𝟐𝒏 + 𝟑)𝟐 − 𝟓 = 𝟒𝒏𝟐 + 𝟏𝟐𝒏 + 𝟗 − 𝟓 = 𝟒𝒏𝟐 + 𝟏𝟐𝒏 + 𝟒 = 𝟒(𝒏𝟐 + 𝟑𝒏 + 𝟏) [1] Since (𝟐𝒏 + 𝟑)𝟐 − 𝟓 = 𝟒(𝒏𝟐 + 𝟑𝒏 + 𝟏), which can be divisible by 4, hence (𝟐𝒏 + 𝟑)𝟐 − 𝟓 is a multiple of 4 for all integer values of n. (shown) [1] 3 In the figure, five numbers are shown below to form a pattern. 5, 12, 19, 26, 33 Aaron says that 112 is a term in the pattern. Is he correct? Explain your answer. [1] No, he is not correct. 7n – 2 = 112, n = 𝟏𝟏𝟒 𝟕 . Since the value of n is not an integer, hence 112 is not a term in the pattern. 4 Let x be the number of minutes a pump can generate water out. Solve the equation 3𝑥2 − 15𝑥 − 100 = 0. Explain why one of your solutions is not acceptable. [1] x = 8.79 or x = -3.79. Since the number of minutes cannot be negative, hence x = 8.79.

Vectors 1 (a) Explain why AB and OC are parallel. [1] Since 𝑶𝑪⃗⃗⃗⃗⃗⃗ = 2𝑨𝑩⃗⃗⃗⃗⃗⃗ , hence AB and OC are parallel. (b) Explain why points O, P and B are collinear. [2] 𝑶𝑷⃗⃗⃗⃗⃗⃗ = 2a – b and 𝑷𝑩⃗⃗⃗⃗⃗⃗ = 2(2a – b) [1] Since 𝑷𝑩⃗⃗⃗⃗⃗⃗ = 2𝑶𝑷⃗⃗⃗⃗⃗⃗ , hence points O, P and B are collinear. [1]

Circle properties 1 Explain why a circle can be drawn with points P, Q, R and S? [1] Angle PQS + angle PRS = 131° + 49° = 180° By converse of angle at opposite segment, a circle can be drawn with points P, Q, R and S.

2 (O level 2013 P1 Qn 13 (a)) ABC is a tangent to the circle at point B. Explain why BD is not a diameter of the circle. [1] Angle DBC = 180° - 53° - 39° = 88° (∠ sum of ∆) Since angle DBC ≠ 90°, by converse of radius ⊥ tangent, BD is not a diameter of the circle.