The answer
(a) \(BD = 54\) m (shown)
(b) \(3177\) m²
O-Level E-Math 2020 Paper 1 Question 22 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 22 of the O-Level E-Math 2020 Paper 1. It tests similar right triangles, in the Mensuration / surveys area. It is worth 5 marks: (a) 1, (b) 4. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) \(BG \perp AD\) (since \(\angle DBG = 90^{\circ}\)), so triangles \(ABG\) and \(GBD\) are both right-angled at \(B\). With \(\angle BAG = \angle BGD\), the triangles are similar, giving \(\dfrac{BD}{BG} = \dfrac{BG}{AB}\). Hence \(BD = \dfrac{BG^2}{AB} = \dfrac{36^2}{24} = \dfrac{1296}{24} = 54\) m.
(b) Working along the line, \(AC = AB + BC = 24 + 37 = 61\) and (from part (a)) \(CD = BD - BC = 54 - 37 = 17\), while \(AD = AB + BD = 24 + 54 = 78\). *Find the missing offset \(CF\) from the given trapezium:* \(ACFE\) has parallel offsets \(EA = 30\) and \(CF\) with width \(AC = 61\), so \(\tfrac12(30 + CF)(61) = 1586 \Rightarrow 30 + CF = 52 \Rightarrow CF = 22\). Now add the two remaining pieces. Triangle \(CDF = \tfrac12 \times CD \times CF = \tfrac12 \times 17 \times 22 = 187\). The whole right side \(ADG\) is a triangle with base \(AD = 78\) and height \(BG = 36\): area \(= \tfrac12 \times 78 \times 36 = 1404\). Total area \(= 1586 + 187 + 1404 = 3177\) m². *(Check by the shoelace formula on \(A(0,0),E(-30,0),F(-22,61),D(0,78),G(36,24)\): area \(= 3177\) m².)*
Answer: (a) \(BD = 54\) m (shown)
(b) \(3177\) m²
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a similar right triangles question from Mensuration / surveys, worth 5 marks: (a) 1, (b) 4.
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