Question

5.
In figure 3.81, seg EF is a diameter
and seg DF is a tangent segment.
The radius of the circle is r.
Prove that, DE x GE = 4r2​

Answer 1

Step-by-step explanation:

Given:circle with centre H

EF is a diameter

Radius =r

Seg DF is a tangent segment

To prove:DE×GE=4r2

Proof:EF=2r........(diameter=2r)...(#)

By tangent secant segment theorem

DF2=DG×DE......(Equation 1)

DG+GE=DE......(D-G-E)

GE=DE-DG.....(Equation 2)

angle EFD=90°.....(BY TANGENT THEOREM)

In ∆EFD,angle EFD=90°

By pythagoras theorem

DE2=DF2+EF2

=DF2+(2r)2.......(from#)

=DF2+4r2

DE2-DF2=4r2

DE2-DG×DE=4r2............(from eq 1)

DE(DE-DG)=4r2

DE×GE=4r2

hence proved