Question

In the given figure
(1) m(arc CE) = 54°, m(arc BD) = 23°, find measure of ∠CAE.
(2) If AB = 4.2, BC = 5.4, AE = 12.0, find AD
(3) If AB = 3.6, AC = 9.0, AD = 5.4, find AE

Answer 1

(1)
If two lines containing chords of a circle intersect each other outside the circle, then the measure of angle between them is half the difference in measures of the arcs intercepted by the angle.

∴ ∠CAE = $\frac{1}{2}$[m(arc CE) − m(arc BD)] = $\frac{1}{2}\times \left(54°-23°\right)=\frac{1}{2}\times 31°$ = 15.5º

(2)
AC × AB = AE × AD (Theorem of external division of chords)

⇒ (AB + BC) × AB = AE × AD

⇒ (4.2 + 5.4) × 4.2 = 12 × AD

⇒ AD = $\frac{9.6\times 4.2}{12}$ = 3.36

(3)
AC × AB = AE × AD (Theorem of external division of chords)

⇒ 9 × 3.6 = AE × 5.4

⇒ AE = $\frac{9\times 3.6}{5.4}$ = 6