IN A TRIANGLE ABC, DE IS PAPRALLEL TO BC. IF AD:DB=3:5, THEN FIND ar(DFE)/ar(CFB)

Asked by smruti2002ranjan | 21st Sep, 2016, 11:20: PM

Expert Answer:

begin mathsize 14px style In space triangle ABC comma space DE space parallel to BC
rightwards double arrow angle ADE space equals space space ABC space and space angle AED space equals space angle ACB space left parenthesis corresponding space angles right parenthesis
Thus comma space in space triangles space ADE space and space ABC comma
angle straight A space equals space angle straight A space space space left parenthesis common right parenthesis
angle ADE space equals space angle ABC
angle AED space equals space angle ACB
rightwards double arrow triangle ADE space tilde triangle ABC space space space left parenthesis By space AAA space similarity right parenthesis
rightwards double arrow AD over AB equals DE over BC
Now comma space AD over DB equals 3 over 5
rightwards double arrow DB over AD equals 5 over 3
rightwards double arrow DB over AD plus space 1 equals 5 over 3 space plus space 1
rightwards double arrow fraction numerator DB plus AD over denominator AD end fraction equals 8 over 3
rightwards double arrow AB over AD equals 8 over 3
rightwards double arrow AD over AB equals 3 over 8
rightwards double arrow DE over BC equals 3 over 8
In space triangle DFE space and space triangle CFB comma
angle 1 space equals space angle 3 space space space space space left parenthesis Alternate space interior space angles right parenthesis
angle 2 space equals space angle 4 space space space left parenthesis Vertically space opposite space angles right parenthesis
rightwards double arrow triangle DFE space tilde space triangle CFB space space space left parenthesis By space AA space similarity right parenthesis
rightwards double arrow fraction numerator Ar left parenthesis triangle DFE right parenthesis over denominator Ar left parenthesis triangle CFB right parenthesis end fraction equals DE squared over BC squared equals open parentheses 3 over 8 close parentheses squared equals 9 over 64
end style

Answered by Rashmi Khot | 22nd Sep, 2016, 10:16: AM