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) m(arc CE) = 54°, m(arc BD) = 23°

∠CAE has its vertex outside the circle and intercept and CE and BD.

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 = `1/2` × [m(arc CE) - m(arc BD)]

∴ ∠CAE = `1/2` × [54 - 23]

∴ ∠CAE = `1/2` × 31

∴ ∠CAE = 15.5º

(2) AB = 4.2, BC = 5.4, AE = 12 ...(Given)

AC = AB + BC       ...(A - B - C)

∴ AC = 4.2 + 5.4

∴ AC = 9.6

Chord CB and chord ED intersect at point A outside the circle.

∴ by theorem of external division of chords.

AB × AC = AD × AE

∴ 4.2 × 9.6 = AD × 12

∴ AD = `(4.2×9.6)/12`

∴ AD = `(4.2 × 9.6 × 100)/(12 × 100)`

∴ AD = `(42 × 96)/(1200)`

∴ AD = `(42 × 8)/(100)`

∴ AD = 3.36

(3) AB = 3.6, AC = 9 cm and AD = 5.4        ...(Given )

chords CB and ED intersect at point A outside the circle.

∴ by theorem of external division of chords,

AB × AC = AD × AE

∴ 3.6 × 9 = 5.4 × AE

∴ AE = `(3.6 × 9)/5.4`

∴  AE = `(3.6 × 9 × 10)/(5.4 × 10)`

∴ AE = `(36 × 9)/54`

∴ AE = 6