Q : 8       Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and
                F respectively. Prove that the angles of the triangle DEF are    \small 90^{\circ}-\frac{1}{2}C,  \small 90^{\circ}-\frac{1}{2}B and \small 90^{\circ}-\frac{1}{2}A

Answers (1)

Given :   Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively.

To prove :  the angles of the triangle DEF are    \small 90^{\circ}-\frac{1}{2}C,    \small 90^{\circ}-\frac{1}{2}B and \small 90^{\circ}-\frac{1}{2}A

Proof : 

          

\angle1 and \angle3 are angles in same segment.therefore,

          \angle1 = \angle3 ................1(angles in same segment are equal )

and    \angle2 = \angle4 ..................2

Adding 1 and 2,we have 

         \angle1+\angle2=\angle3+\angle4

\Rightarrow \angle D=\frac{1}{2}\angle B+\frac{1}{2}\angle C,

\Rightarrow \angle D=\frac{1}{2}(\angle B+\angle C)

\Rightarrow \angle D=\frac{1}{2}(180 \degree+\angle C)

and  \Rightarrow \angle D=\frac{1}{2}(180 \degree-\angle A)

        \Rightarrow \angle D=90 \degree-\frac{1}{2}\angle A

Similarly,  \Rightarrow \angle E=90 \degree-\frac{1}{2}\angle B     and    \angle F=90 \degree-\frac{1}{2}\angle C

 

 

 

 

 

 

 

 

 

Preparation Products

NEET Foundation 2022

NEET Foundation 2022.

₹ 14999/-
Buy Now
Biology Foundation for Class 10

Biology Foundation for Class 10.

₹ 999/- ₹ 499/-
Buy Now
Knockout JEE Main April 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout NEET May 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout JEE Main May 2021

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 22999/- ₹ 9999/-
Buy Now
Boost your Preparation for JEE Main with our Foundation Course
 
Exams
Articles
Questions