The perimeter of a rhombus is 52cm and one of the diagonal is 10cm.find the length of the other diagonal and the area of rhombus
Answers
Perimeter of a rhombus = perimeter of a square =4×side =52cm
=4×side = 52
Side=52÷4= 13 cm
Note: We know that the diagonals of a rhombus bisect each other at right angles.
Length of the one diagonal = 10cm.
Then the length of the other diagonal will be:
= 24cm.
We know that Area of the rhombus = 1/2 * product of the diagonals
= 1/2 * 10 * 24
= 24 * 5
= 120cm^2.
Steps are given in the attached image
Answer:
The length of the other diagonal of rhombus is 24 cm.
The area of the rhombus is 120 cm².
Step-by-step-explanation:
In figure, ⎕ ABCD is a rhombus.
AC = 10 cm - - [ Given ]
Now, we know that,
Perimeter of rhombus = 4 × side
P ( ⎕ ABCD ) = 4 × side
→ 52 = 4 × side
→ side = 52 / 4
→ Side = 13 cm
∴ AB = BC = CD = AD = 13 cm - - [ All sides of rhombus are congruent. ] ( 1 )
Now,
AC = 10 cm - - [ Given ]
∴ AO = OC = 1 / 2 × AC - - [ Diagonals of rhombus bisect each other. ]
→ AO = OC = 1 / 2 × 10
→ AO = OC = 5 cm - - ( 2 )
Now, in Δ AOD, ∠ AOD = 90°. - - [ Diaonlas of rhombus are perpendicular bisectors of each other. ]
AD² = AO² + OD² - - - [ Pythagoras theorem ]
→ ( 13 )² = ( 5 )² + OD² - - [ From ( 1 ) & ( 2 ) ]
→ 169 = 25 + OD²
→ OD² = 169 - 25
→ OD² = 144
→ OD = 12 cm - - - [ Taking square roots ] ( 3 )
Now,
OD = OB = 1 / 2 × BD - - [ Diagonals of rhombus bisect each other. ]
→ BD = 2 × OD
→ BD = 2 × 12 - - [ From ( 3 ) ]
→ BD = 24 cm - - ( 4 )
Now, we know that,
Area of rhombus = Product of diagonals / 2
∴ A ( ⎕ ABCD ) = ( AC × BD ) / 2
→ A ( ⎕ ABCD ) = ( 10 × 24 ) / 2 - - [ From ( 1 ) & ( 4 ) ]
→ A ( ⎕ ABCD ) = 240 / 2
→ A ( ⎕ ABCD ) = 120 cm².
∴ The length of the other diagonal of rhombus is 24 cm.
The area of the rhombus is 120 cm².
