Advertisements
Advertisements
प्रश्न
A rope of length 42 m is given. Find the largest area of the triangle formed by this rope and find the dimensions of the triangle so formed
उत्तर
Let a, b, c be the lengths of the sides of the triangle.
Given the perimeter of the triangle
2s = a + b + c = 42 m ......(1)
For a fixed perimeter 2s, the area of a triangle is maximum
when a = b = c.
(1) ⇒ a + a + a = 42
3a = 42 ⇒ a = `42/3`
a = 14 m
∴ a = b = c = 14 m
The maximum area of the triangle ∆ = `8^2/(3sqrt(3))`
= `21^2/(3sqrt(3))`
= `(21 xx 21)/(3sqrt(3))`
= `(7 xx 3 xx 7)/sqrt(3)`
= `49sqrt(3)` sq.m.
∴ The dimensions of the triangle are 14 m, 14 m, 14 m.
Maximum area = `49sqrt(3)` sq.m.
APPEARS IN
संबंधित प्रश्न
In a ∆ABC, if `sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C"))` prove that a2, b2, C2 are in Arithmetic Progression
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 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
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 usual notation a ΔABC, if A, A1, A2, A3 be the area of the in-circle and ex-circles, then `1/sqrt(A_1) + 1/sqrt(A_2) + 1/sqrt(A_3)` is equal to ______.
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 ______.
If in a ΔABC, the altitudes from the vertices A, B, C on opposite sides are in H.P, then sin A, sin B, sin C are in ______
