Question

9. (a) The diagram below, not drawn to scale, shows a circle, where two lines intersect at point C. The points A, B, D and I lie on the circumference of the circle Note that ABDE is a right-angled triangle and BD is the diameter of the circle. A 66° D 78° C E B Determine, giving a reason for your answer, (1) ВСЕ 121 (ii) BDE 121 (iii) DBE 121

Answer 1

(i)

The angle 78° is supplementary to the angle BCE. Then we have:

`\begin{gathered} 78\degree+B\hat{C}E=180\degree \\ B\hat{C}E=180\degree-78\degree \\ B\hat{C}E=102\degree \end{gathered}`

(ii)

When the vertex of a angle formed by two segments is located on the circle, the corresponding arc formed by the two segments is the double of the angle. Then we have:

`\begin{gathered} B\hat{A}E=B\hat{D}E=\frac{arc\text{ BR}}{2} \\ B\hat{A}E=66\degree \\ \therefore B\hat{D}E=66\degree \end{gathered}`

(iii)

Since BDE is a right triangle, we have:

`\begin{gathered} D\hat{B}E+B\hat{D}E+90\degree=180\degree \\ D\hat{B}E+66\degree+90\degree=180\degree \\ D\hat{B}E=180\degree-90\degree-66\degree \\ D\hat{B}E=24\degree \end{gathered}`