In the 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) `/_ CAE = 15.5^(@) , ` (2) `AD = 3.36` and (3) AE = 6 .

Explanation

(1) m (arc CE ) `= 54^(@)` and m (arc BD ) `= 23^(@)` …(Given )
`/_CAE ` has its vertex outside the circle and intercept and CE and BD.
`:. /_CAE = (1)/(2) xx [ m (arc CE ) - m ( arc BD ) ]`
`:. /_CAE = (1)?(2) [ 54 - 23]`
`:. /_ CAE = (1)/(2) [ 31] `
` :. /_CAE = 15.5^(@0`.
(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 xx AC = AD xx AE `
`:. 4.2 xx 9.6 = AD xx 12 `
` :. AD = ( 4.2 xx 9. 6 )/( 12 ) `
`:. AD = ( 4.2 xx 9.6 xx 100)/( 12 xx 100)`
` :. AD = ( 4.2 xx 96)/9 12 xx 100`
`:. AD = ( 336 )/( 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 xxAC = AD xx AE `
`:. 3.6 xx 9 = 5.4 xx AE `
`:. AE = ( 3.6 xx 9 )/(5.4 ) `
` :. AE = ( 3.6 xx 9 xx 10)/( 5.4 xx 10 )`
`:. AE = ( 36 xx 9 )/( 54)`
`:. AE = 6`