In a right ∆ABC (angle B=90°), AB=5cm and BC=12cm. CD and AE are angle bisectors of angle C and A respectively and they intersect at I. Find thearea of ∆DIE.
Answers
Given : In a right ∆ABC (angle B=90°), AB=5cm and BC=12cm. CD and AE are angle bisectors of angle C and A respectively and they intersect at I.
To find : area of ∆DIE
Solution:
AB = 5 cm
BC = 12 cm
AC² = AB² + BC² = 5² + 12² = 13²²
=> AC = 13
Area of ΔABC = (1/2) AB * BC = (1/2) * 5 * 12 = 30 cm²
Area of ΔABC = (1/2)(AB + BC + AC) * r = (1/2)*(5 + 12 + 13)r = 15r
( r = inradius)
15r = 30
=> r = 2
CD is angle bisector
=> AC/AD = BC/BD
=> 13/(5 - BD) = 12/BD
=> 13BD = 60 - 12BD
=> BD = 12/5
AE is angle bisector
=> AB/BE = AC/CE
=> 5/BE = 13/(12 - BE)
=> 60 - 5BE = 13BE
=> BE = 10/3
Area of ΔDIE = Area of Δ DBI + Area of ΔEBI - Area of Δ DBE
= (1/2) BD * r + (1/2)BE * r - (1/2)BD * BE
= (1/2) (12/5) * 2 + (1/2)(10/3)*2 - (1/2)(12/5)(10/3)
= 12/5 + 10/3 - 4
= 86/15 - 4
= 26/15
area of ∆DIE = 26/15 cm²
Learn More:
In a triangle abc, ad is angle bisector of a and ab : ac = 3 : 4. If the ...
https://brainly.in/question/11320075
ind the length of angle bisector of triangle abc where a = 6 CM B = 6 ...
https://brainly.in/question/8651539