Question

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

Answer 1

Answer:

a) 78°

b) 39°

Step-by-step 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.

$m\angle CAE = \frac{1}{2} \times m\widehat{(AC)} \\\\\therefore 39\degree = \frac{1}{2} \times m\widehat{(AC)} \\\\\therefore m\widehat{(AC)} = 39\degree\times 2\\\red{\bold{\therefore m\widehat{(AC)} = 78\degree}}$

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

$\therefore m\angle AOC = m\widehat{(AC)}\\\huge \purple {\boxed {\therefore m\angle AOC = 78\degree}}$

Next, by inscribed angle theorem:

$m\angle CBA = \frac{1}{2}\times m\widehat{(AC)}\\\\\therefore m\angle CBA = \frac{1}{2}\times 78\degree \\\\\therefore m\angle CBA =39\degree \\\\\huge \orange{\boxed {\therefore m\angle OBA = 39\degree}}\\... (\because C-O-B)$