The area of the rhombus is 6084 cm² and one of its diagonals is 52 cm long . Find the length of the other diagonal, also find its perimeter.
Answers
Given data :
➜ Area of the rhombus = 6084 cm²
➜ Length of the diagonal = 52 cm
Solution : Let, first and second diagonal of rhombus be AC and be BD respectively. {according to figure}
Hence, AC = 52 cm and BD = ?
Now, by formula;
➜ Area of rhombus = {product of diagonals}/2
➜ Area of rhombus = {AC * BD}/2
➜ 6084 = {52 * BD}/2
➜ 52 * BD = 6084 * 2
➜ 52 * BD = 12168
➜ BD = 12168/52
➜ BD = 234 cm
Here, we know, that, In a rhombus, diagonals bisect each other at right angles and all sides of the rhombus are equal. Hence, AO = OC and BO = OD.
Now,
➜ AC = AO + OC
➜ AC = AO + AO
➜ AC = 2 * AO
➜ AO = AC/2
➜ AO = 52/2
➜ AO = 26 cm
Similarly
➜ BD = BO + OD
➜ 234 = BO + BO
➜ BD = 2 * BO
➜ BO = BD/2
➜ BO = 234/2
➜ BO = 117 cm
Now, by Pythagoras theorem,
➜ {Hypo}² = {AO}² + {BO}²
Where, Hypotenuse = AB
➜ {AB}² = {AO}² + {BO}²
➜ {AB}² = {26}² + {117}²
➜ {AB}² = 676 + 13689
➜ {AB}² = 14365
➜ AB = √14365
➜ AB = 13√85 cm or 119.8540 cm
Hence, length of each side of rhombus is 13√85 cm.
Now, by formula of perimeter of rhombus,
➜ Perimeter of rhombus = 4 * side
➜ Perimeter of rhombus = 4 * 13√85
➜ Perimeter of rhombus = 52√85 cm
{52√85 cm = 479.4263 cm}
Answer : Hence, the length of the diagonal of the rhombus is 234 cm and perimeter of the rhombus is 52√85 cm.
More info : Properties of Rhombus;
All sides of the rhombus are equal.
The opposite sides of a rhombus are parallel.
Opposite angles of a rhombus are equal.
The sum of two adjacent angles is equal to 180 degrees.
The sum of interior angles of a rhombus add up to 360 degrees.
The opposite angles of a rhombus are equal to each other.
In a rhombus, diagonals bisect each other at right angles.