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If the perimeter of a rhombus is 80 cm and one of its diagonal is 24 cm, then what is the area (in cm2) of the rhombus? (SSC Sub. Ins. 2017)
- a)218
- b)192
- c)384
- d)768
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
If the perimeter of a rhombus is 80 cm and one of its diagonal is 24 c...
By Pythagorean theorem,
(20)2 = (12)2 + (x2)
∴ x2 = 400 – 144 = 256
∴ x = √256 = 16 cm
∴ Diagonal of rhombus = 2x = 2 × 16 = 32 cm and other diagonal = 24 cm
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If the perimeter of a rhombus is 80 cm and one of its diagonal is 24 c...
Understanding the Problem
To find the area of the rhombus, we need to use the information provided: the perimeter and one diagonal.
Step 1: Find the Length of the Sides
- The perimeter of a rhombus is given by the formula:
Perimeter = 4 × Side
- Given that the perimeter is 80 cm:
4 × Side = 80 cm
- Thus, the length of each side is:
Side = 80 cm / 4 = 20 cm
Step 2: Using the Diagonal
- One diagonal (d1) is given as 24 cm.
- Let the other diagonal be d2.
- The diagonals of a rhombus bisect each other at right angles.
Step 3: Applying the Pythagorean Theorem
- Each half of the diagonals forms a right triangle with the side of the rhombus:
(d1/2)² + (d2/2)² = Side²
- Substituting the known values:
(24/2)² + (d2/2)² = 20²
- This simplifies to:
12² + (d2/2)² = 400
- Therefore:
144 + (d2/2)² = 400
(d2/2)² = 256
d2/2 = 16
d2 = 32 cm
Step 4: Calculating the Area
- The area (A) of a rhombus can be calculated using the formula:
A = (d1 × d2) / 2
- Substituting the values of the diagonals:
A = (24 cm × 32 cm) / 2
= 768 cm² / 2
= 384 cm²
Conclusion
The area of the rhombus is 384 cm², confirming that the correct answer is option 'C'.
To find the area of the rhombus, we need to use the information provided: the perimeter and one diagonal.
Step 1: Find the Length of the Sides
- The perimeter of a rhombus is given by the formula:
Perimeter = 4 × Side
- Given that the perimeter is 80 cm:
4 × Side = 80 cm
- Thus, the length of each side is:
Side = 80 cm / 4 = 20 cm
Step 2: Using the Diagonal
- One diagonal (d1) is given as 24 cm.
- Let the other diagonal be d2.
- The diagonals of a rhombus bisect each other at right angles.
Step 3: Applying the Pythagorean Theorem
- Each half of the diagonals forms a right triangle with the side of the rhombus:
(d1/2)² + (d2/2)² = Side²
- Substituting the known values:
(24/2)² + (d2/2)² = 20²
- This simplifies to:
12² + (d2/2)² = 400
- Therefore:
144 + (d2/2)² = 400
(d2/2)² = 256
d2/2 = 16
d2 = 32 cm
Step 4: Calculating the Area
- The area (A) of a rhombus can be calculated using the formula:
A = (d1 × d2) / 2
- Substituting the values of the diagonals:
A = (24 cm × 32 cm) / 2
= 768 cm² / 2
= 384 cm²
Conclusion
The area of the rhombus is 384 cm², confirming that the correct answer is option 'C'.
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If the perimeter of a rhombus is 80 cm and one of its diagonal is 24 cm, then what is the area (in cm2) of the rhombus? (SSC Sub. Ins. 2017)a)218b)192c)384d)768Correct answer is option 'C'. Can you explain this answer?
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