Question

8. In the diagram below, angle CAE = 39 °. Find the following
:) AOC
b) OBA​

Answer 1

a) 78°

b) 39°

Explanation:

In circle with center O, DE is tangent at point A and AC is chord (or secant if extended both sides).

Therefore by tangent secant theorem.

`m\angle CAE = (1)/(2) * m\widehat{(AC)} \\\\</p><p>\therefore 39\degree = (1)/(2) * m\widehat{(AC)} \\\\</p><p>\therefore m\widehat{(AC)} = 39\degree* 2\\</p><p>\red{\bold{\therefore m\widehat{(AC)} = 78\degree}}\\`

Since, measure of central angle of a circle is equal to the measure of its corresponding minor arc.

`\therefore m\angle AOC = m\widehat{(AC)}\\</p><p>\huge \purple {\boxed {\therefore m\angle AOC = 78\degree}} \\`

Next, by inscribed angle theorem:

`m\angle CBA = (1)/(2)* m\widehat{(AC)}\\\\</p><p>\therefore m\angle CBA = (1)/(2)* 78\degree \\\\</p><p>\therefore m\angle CBA =39\degree \\\\</p><p>\huge \orange{\boxed {\therefore m\angle OBA = 39\degree}}\\... (\because C-O-B) \\`