Advertisements
Advertisements
प्रश्न
In a ∆ABC, ∠A = 60°. Prove that b + c = `2"a" cos (("B" - "C")/2)`
उत्तर
Given ∠A = 60°
A + B + C = 180°
60° + B + C = 180°
B + C = 180° – 60° = 120°
We have `"a"/sin"A" = "b"/sin"B" = "c"/sin"C"` = 2R
`"a"/sin"A"` = 2R ⇒ a = 2R sin A
`"b"/sin"B"` = 2R ⇒ b = 2R sin B
`"c"/sin"C"` = 2R ⇒ c = 2R sin C
b + c = 2R sin B + 2R sin C
= 2R (sin B + sin C)
= `2"R" * 2sin (("B" + "C")/2) * cos (("B" - "C")/2)`
= `4"R" * sin(120^circ/2) * cos (("B" - "C")/2)`
= `4"R" * sin 60^circ * cos (("B" - "C")/2)`
= `2 * 2"R" * sin"A" * cos (("B" - "C")/2)`
b + c = `2"A" cos (("B" - "C")/2)`
APPEARS IN
संबंधित प्रश्न
The angles of a triangle ABC, are in Arithmetic Progression and if b : c = `sqrt(3) : sqrt(2)`, find ∠A
In a ∆ABC, if cos C = `sin "A"/(2sin"B")` show that the triangle is isosceles
In an ∆ABC, prove that a cos A + b cos B + c cos C = 2a sin B sin C
In a ∆ABC, prove the following, a(cos B + cos C) = `2("b" + "c") sin^2 "A"/2`
In a ∆ABC, prove the following, `("a"^2 - "c"^2)/"b"^2 = (sin ("A" - "C"))/(sin("A" + "C"))`
In a ∆ABC, prove the following, `("a"sin("B" - "C"))/("b"^2 - "c"^2) = ("b"sin("C" - "A"))/("c"^2 - "a"^2) = ("c"sin("A" - "B"))/("a"^2 - "b"^2)`
In a ∆ABC, prove the following, `("a"+ "b")/("a" - "b") = tan(("A" + "B")/2) cot(("A" - "B")/2)`
In a ∆ABC, prove that (a2 – b2 + c2) tan B = (a2 + b2 – c2) tan C
An Engineer has to develop a triangular shaped park with a perimeter 120 m in a village. The park to be developed must be of maximum area. Find out the dimensions of the park
Derive Projection formula from Law of sines
Derive Projection formula from Law of cosines
Choose the correct alternative:
In a ∆ABC, if
(i) `sin "A"/2 sin "B"/2 sin "C"/2 > 0`
(ii) sin A sin B sin C > 0 then
A circle touches two of the smaller sides of a ΔABC (a < b < c) and has its centre on the greatest side. Then the radius of the circle is ______.
In a ΔABC, let BC = 3. D is a point on BC such that BD = 2, Then the value of AB2 + 2AC2 – 3AD2 is ______.
In an equilateral triangle of side `2sqrt(3)` cm, the circum radius is ______.
Let a, b and c be the length of sides of a triangle ABC such that `(a + b)/7 = (b + c)/8 = (c + a)/9`. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of `R/r` is equal to ______.
