In a triangle the angles A , B , C are in A.P. Show that 2 cos A - C 2 = a + c a 2 - a c + c 2 .

Given that angles A , B , C of ∆ ABC are in A . P . ∴ 2 B = A + C Sum of all angles of ∆ ABC is 180 ° . ⇒ A + B + C = 180 ° ⇒ 3 B = 180 ° ⇒ B = 60 ° ⇒ cos B = cos 60 ° We know that, by cosine rule cos B = a 2 + c 2 - b 2 2 a c ⇒ a 2 + c 2 - b 2 2 a c = 1 2 ⇒ a 2 + c 2 - a c = b 2 (i) We know that, by sine rule a sin A = b sin B = c sin C = k ( say ) ⇒ k sinA = a , k sinB = b , k sinC = c ∴ R . H . S = a + c a 2 - a c + c 2 = a + c b 2 = a + c b = k sinA + k sinC k sinB = sinA + sinC sinB = 2 sin A + C 2 cos A - C 2 sin B = 2 sin B cos A - C 2 sin B [ ∵ A + C = 2 B ] = 2 cos A - C 2 = L · H . S Hence, proved.

In a triangle A B C , a : b : c = 3 : 1 : 1 , then the triangle is

acute angled triagle, which is not isosceles

E M B I B E

Beta Question Bank for Engineering: Mathematics>Chapter 34 - Properties of Triangle>EXERCISE-4>Q 1

HARD

JEE Main/Advance

IMPORTANT

Earn 100

$A$ is a point on the circumference of a circle. Chords $A B$ and $A C$ divide the area of the circle into three equal parts. If the angle $B A C$ is the root of the equation, $f (x) = 0$ then find $f (x)$ .

Important Questions on Properties of Triangle

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 34 - Properties of Triangle>EXERCISE-1>Q 2

A

and

B

are two points

30 m

apart in a line on the horizontal plane through the foot of a tower lying on opposite sides of the tower. If the distance of the top of the tower from