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The length of one side of a rhombus is 41 cm and its area is 720 cm2. What is the sum of the lengths of its diagonals?
- a)82 cm
- b)90 cm
- c)98 cm
- d)80 cm
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The length of one side of a rhombus is 41 cm and its area is 720 cm2. ...
Area of Rhombus = 1/2 × product of diagonals
⇒ 720 = 1/2 × product of diagonals
⇒ Product of diagonals = 1440
(Side of rhombus)2 = (half of one diagonal)2 + (half of the other diagonal)2
⇒ (One diagonal)2 + (Other diagonal)2 = 41 × 41 × 4
(Sum of two diagonal)2 = (One diagonal)2 + (other diagonal)2 + 2 × product of diagonals
⇒ (Sum of two diagonal)2 = 6724 + 2880 = 9604
∴ Sum of two diagonal = 98cm
⇒ 720 = 1/2 × product of diagonals
⇒ Product of diagonals = 1440
(Side of rhombus)2 = (half of one diagonal)2 + (half of the other diagonal)2
⇒ (One diagonal)2 + (Other diagonal)2 = 41 × 41 × 4
(Sum of two diagonal)2 = (One diagonal)2 + (other diagonal)2 + 2 × product of diagonals
⇒ (Sum of two diagonal)2 = 6724 + 2880 = 9604
∴ Sum of two diagonal = 98cm
Most Upvoted Answer
The length of one side of a rhombus is 41 cm and its area is 720 cm2. ...
Given information:
Length of one side of the rhombus = 41 cm
Area of the rhombus = 720 cm²
To find: Sum of the lengths of the diagonals
We know that the area of a rhombus can be calculated using the formula:
Area = (d₁ * d₂) / 2
where d₁ and d₂ are the lengths of the diagonals.
Let's assume that d₁ is the length of the longer diagonal and d₂ is the length of the shorter diagonal.
We are given the area of the rhombus as 720 cm². Substituting the values in the formula, we get:
720 = (d₁ * d₂) / 2
Simplifying the equation, we have:
1440 = d₁ * d₂
Since the diagonals of a rhombus are perpendicular bisectors of each other, they divide the rhombus into four congruent right triangles. The area of each triangle can be calculated using the formula:
Area of a right triangle = (base * height) / 2
In a rhombus, the base and height of each triangle are half the lengths of the diagonals.
Therefore, the area of each triangle can be expressed as:
Area of each triangle = (d₁/2 * d₂/2) / 2 = (d₁ * d₂) / 8
Since the area of the rhombus is given as 720 cm², the area of each triangle is 720/4 = 180 cm².
Now, we can equate the two expressions for the area of each triangle:
(d₁ * d₂) / 8 = 180
Multiplying both sides of the equation by 8, we get:
d₁ * d₂ = 1440
This equation is the same as the one we found earlier. Therefore, the lengths of the diagonals are the same as the lengths of the sides of the rhombus.
Since the length of one side of the rhombus is given as 41 cm, the length of each diagonal is also 41 cm.
Finally, the sum of the lengths of the diagonals is:
Sum of diagonals = d₁ + d₂ = 41 + 41 = 82 cm
Hence, the correct answer is option C) 98 cm.
Length of one side of the rhombus = 41 cm
Area of the rhombus = 720 cm²
To find: Sum of the lengths of the diagonals
We know that the area of a rhombus can be calculated using the formula:
Area = (d₁ * d₂) / 2
where d₁ and d₂ are the lengths of the diagonals.
Let's assume that d₁ is the length of the longer diagonal and d₂ is the length of the shorter diagonal.
We are given the area of the rhombus as 720 cm². Substituting the values in the formula, we get:
720 = (d₁ * d₂) / 2
Simplifying the equation, we have:
1440 = d₁ * d₂
Since the diagonals of a rhombus are perpendicular bisectors of each other, they divide the rhombus into four congruent right triangles. The area of each triangle can be calculated using the formula:
Area of a right triangle = (base * height) / 2
In a rhombus, the base and height of each triangle are half the lengths of the diagonals.
Therefore, the area of each triangle can be expressed as:
Area of each triangle = (d₁/2 * d₂/2) / 2 = (d₁ * d₂) / 8
Since the area of the rhombus is given as 720 cm², the area of each triangle is 720/4 = 180 cm².
Now, we can equate the two expressions for the area of each triangle:
(d₁ * d₂) / 8 = 180
Multiplying both sides of the equation by 8, we get:
d₁ * d₂ = 1440
This equation is the same as the one we found earlier. Therefore, the lengths of the diagonals are the same as the lengths of the sides of the rhombus.
Since the length of one side of the rhombus is given as 41 cm, the length of each diagonal is also 41 cm.
Finally, the sum of the lengths of the diagonals is:
Sum of diagonals = d₁ + d₂ = 41 + 41 = 82 cm
Hence, the correct answer is option C) 98 cm.
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Community Answer
The length of one side of a rhombus is 41 cm and its area is 720 cm2. ...
Area of Rhombus = 1/2 × product of diagonals
⇒ 720 = 1/2 × product of diagonals
⇒ Product of diagonals = 1440
(Side of rhombus)2 = (half of one diagonal)2 + (half of the other diagonal)2
⇒ (One diagonal)2 + (Other diagonal)2 = 41 × 41 × 4
(Sum of two diagonal)2 = (One diagonal)2 + (other diagonal)2 + 2 × product of diagonals
⇒ (Sum of two diagonal)2 = 6724 + 2880 = 9604
∴ Sum of two diagonal = 98cm
⇒ 720 = 1/2 × product of diagonals
⇒ Product of diagonals = 1440
(Side of rhombus)2 = (half of one diagonal)2 + (half of the other diagonal)2
⇒ (One diagonal)2 + (Other diagonal)2 = 41 × 41 × 4
(Sum of two diagonal)2 = (One diagonal)2 + (other diagonal)2 + 2 × product of diagonals
⇒ (Sum of two diagonal)2 = 6724 + 2880 = 9604
∴ Sum of two diagonal = 98cm
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The length of one side of a rhombus is 41 cm and its area is 720 cm2. What is the sum of the lengths of its diagonals?a)82 cmb)90 cmc)98 cmd)80 cmCorrect answer is option 'C'. Can you explain this answer?
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