A community library has a circular stained-glass window of centre \(O\). Two straight wooden struts touch the edge of the window at points \(B\) and \(C\), ...
Question 1 Report
A community library has a circular stained-glass window of centre \(O\). Two straight wooden struts touch the edge of the window at points \(B\) and \(C\), meeting outside the window at point \(A\). Angle \(BOC = 128^\circ\), as shown.
- Give a reason why angle \(OBA = 90^\circ\). (1)
- Work out the size of angle \(BAC\), giving a reason at each stage of your working. (3)
Answer Details
This question uses the circle theorem that a tangent meets a radius at a right angle, and then the angle sum of a quadrilateral, to find an angle between two tangents.
\(OB\) is a radius and \(AB\) is a tangent to the circle at \(B\). A tangent to a circle always meets the radius drawn to the point of contact at \(90^\circ\), so angle \(OBA = 90^\circ\) [1 mark].
By the same tangent-radius reasoning, angle \(OCA = 90^\circ\) as well. The four angles of quadrilateral \(OBAC\) sum to \(360^\circ\), so:
\[\text{angle } BAC = 360^\circ - 90^\circ - 90^\circ - 128^\circ = 52^\circ\][3 marks]
Angle \(BAC\) is the angle actually formed by the two struts at the point where they meet outside the circle; recognising \(OBAC\) as a quadrilateral (rather than trying to work directly in a triangle) is what makes the angle sum available here.
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