Question

in the fig.l and m are two parallel tangents at a and b .the tangent at c makes an interpret de between l and m.prove that angle dfe =90

Answer 1

Given: l and m are two parallel tangents at A and B. The tangent at C makes an intercept DE between l and m.

To Prove: ∠DFE=

Construction: Join AF,FB,FC

Proof:

In triangles ADF and DFC,

DA = DC[ Tangents drawn from an external point are equal in length]

DF = DF [Common]

AF = CF [Radii of the same circle]

∴ΔADF ΔCDF [By SSS congruence]

⇒∠ADF = ∠CDF

⇒∠ADC = 2∠CDF.................(i)

Similarly,

⇒∠CEB = 2∠CEF...................(ii)

Now, ∠ADC+ ∠CEB =

[Sum of Interior angles on the same side of transversal is , as l is parallel to m]

From (i) and (ii), we get,

⇒2∠CDF+ 2∠CEF=

⇒∠CDF+ ∠CEF= .......................(iii)

In ΔDEF

⇒∠DEF+ ∠DFE+∠FDE=

⇒+ ∠DFE= ...............[Using (iii) ]

⇒∠DFE= 90

Answer 2

Given: l and m are two parallel tangents at A and B. The tangent at C makes an intercept DE between l and m.

To Prove: ∠DFE=

Construction: Join AF,FB,FC

Proof:

In triangles ADF and DFC,

DA = DC[ Tangents drawn from an external point are equal in length]

DF = DF [Common]

AF = CF [Radii of the same circle]

∴ΔADF ΔCDF [By SSS congruence]

⇒∠ADF = ∠CDF

⇒∠ADC = 2∠CDF.................(i)

Similarly,

⇒∠CEB = 2∠CEF...................(ii)

Now, ∠ADC+ ∠CEB =

[Sum of Interior angles on the same side of transversal is , as l is parallel to m]

From (i) and (ii), we get,

⇒2∠CDF+ 2∠CEF=

⇒∠CDF+ ∠CEF= .......................(iii)

In ΔDEF

⇒∠DEF+ ∠DFE+∠FDE=

⇒+ ∠DFE= ...............[Using (iii) ]

⇒∠DFE= 90

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