A tram inspector is fixing a new sign at point \(C\) along a route. Stop \(A\) and junction \(B\) are \(40\) m apart, angle \(ABC = 35^\circ\), and \(AC = 2...
Question 1 Report
A tram inspector is fixing a new sign at point \(C\) along a route. Stop \(A\) and junction \(B\) are \(40\) m apart, angle \(ABC = 35^\circ\), and \(AC = 25\) m. Two possible positions for \(C\), labelled \(C_1\) and \(C_2\), are shown below.
- Use the sine rule to find the two possible values of angle \(AC_1B\) and angle \(AC_2B\), each correct to 1 decimal place. (3)
- Hence work out the two corresponding possible values of angle \(BAC\). (2)
- Explain why this information gives two different possible positions for \(C\) rather than one. (1)
Answer Details
This is the ambiguous ("SSA") case of the sine rule: knowing two sides and a non-included angle can give two different triangles, because both an acute angle and its supplement have the same sine.
- The sine rule gives \(\frac{\sin(\angle ACB)}{AB}=\frac{\sin B}{AC}\), so \(\sin(\angle ACB)=\frac{40\sin35^\circ}{25}=0.918\). [1 mark] Taking the inverse sine directly gives the acute solution, angle \(AC_1B=\sin^{-1}(0.918)=66.6^\circ\) (1 d.p.). [1 mark] Since \(\sin\theta=\sin(180^\circ-\theta)\), the obtuse solution is angle \(AC_2B=180^\circ-66.6^\circ=113.4^\circ\). [1 mark]
- Using the angle sum of a triangle for each case: angle \(BAC_1=180^\circ-35^\circ-66.6^\circ=78.4^\circ\). [1 mark] Angle \(BAC_2=180^\circ-35^\circ-113.4^\circ=31.6^\circ\). [1 mark]
- Side \(AC=25\) m is longer than the perpendicular distance from \(A\) to the line through \(B\) but shorter than \(AB=40\) m, so a circular arc of radius \(AC\) centred at \(A\) crosses that line at two distinct points; each crossing gives a valid triangle, which is why two different positions for \(C\) satisfy the given information. [1 mark]
Download The App On Google Playstore
Everything you need to excel in your exams
Personalized AI Learning Chat Assistant
200,000+ Exam Questions Across IGCSE, JAMB, WAEC & NECO
Over 3,900 Lesson Notes
Offline Support - Learn Anytime, Anywhere
Green Bridge Timetable
Literature Summaries & Potential Questions
Track Your Performance & Progress
In-depth Explanations for Comprehensive Learning