Question

The perimeter of a rhombus is 146 cm. One of its diagonals is 55 cm. Then find the length of the other diagonal and the area of the rhombus.

Please tell it fastly using the herons formula​

Answer 1

Answer:

The length of the other diagonal = 48 cm

and its area = 1320 cm²

Step-by-step explanation:

Let the rhombus be ABCD and its diagonal intersect at O

Thus, we know by the Properies of rhombus that,

AB = BC = CD = AD

also, OA = OC and OB = OD

also, ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°

Let AC = 55cm

then, OA + OC = 55 cm

but, OA = OC

So, OA + OA = 55 cm

2 × OA = 55 cm

OA = 55/2 = 27.5 cm

Now, we know that

AB + BC + CD + AD = 146cm

But, AB = BC = CD = AD

so, AB + AB + AB + AB = 146 cm

4 × AB = 146 cm

AB = 146/4 = 73/2 = 36.5 cm

Thus, AB = BC = CD = AD = 36.5 cm

Now,

∠AOD = 90°

Thus, ΔAOD is a right angled triangle

so, By Pythagoras theorem,

AD² = OA² + OD²

36.5² = 27.5² + OD²

OD² = 36.5² - 27.5²

OD² = 1332.25 - 756.25

OD² = 576

OD = √576 = 24 cm

Now, DB = OD + OB

but, OD = OB

DB = OD + OD

DB = 2 × OD

DB = 2 × 24 = 48 cm

Now, area of rhombus = (1/2) × d1 × d2

where d1 and d2 are diagonals

Area = (1/2) × 48 × 55

Area = 1320 cm²

Thus, the length of the other diagonal = 48 cm

and its area = 1320 cm²

Hope it helped and you understood it........All the best