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A rhombus is formed by joining the midpoints of the sides of a unit square.
Q. What is the diameter of the largest circle that can be inscribed within the rhombus?
Q. What is the diameter of the largest circle that can be inscribed within the rhombus?
- a)1/√2
- b)1/2√2
- c)√2
- d)2√2
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A rhombus is formed by joining the midpoints of the sides of a unit sq...
To find the diameter of the largest circle that can be inscribed within the rhombus, we need to find the length of the diagonal of the rhombus.
The sides of the rhombus are formed by joining the midpoints of the sides of the unit square. Since each side of the square has length 1, the sides of the rhombus also have length 1.
The diagonals of a rhombus bisect each other at right angles. Since the sides of the rhombus have length 1, the diagonals are also of length 1.
Now, let's draw the rhombus:
```
A
D-----C
| |
| O |
| |
B-----E
F
```
Let A, B, C, and D be the vertices of the rhombus, and let E be the midpoint of AB and F be the midpoint of BC. The circle with the largest diameter that can be inscribed within the rhombus is tangent to the sides of the rhombus at the midpoints E and F.
Let O be the center of the circle. Since the circle is tangent to the sides of the rhombus at E and F, the radii of the circle are perpendicular to the sides EF and CF.
Since EF is the diagonal of a square with side length 1, EF has length sqrt(2).
Since the diagonals of a rhombus bisect each other at right angles, the length of CF is half the length of the diagonal AC. Since AC has length 1, CF has length 1/2.
Using the Pythagorean theorem, we can find the length of OF:
OF^2 = OC^2 - CF^2
OF^2 = 1/2^2 - 1/2^2
OF^2 = 1/4 - 1/4
OF^2 = 0
This means that OF has length 0, so the circle with the largest diameter that can be inscribed within the rhombus is a point circle. The diameter of this circle is 0.
Therefore, the diameter of the largest circle that can be inscribed within the rhombus is 0.
The sides of the rhombus are formed by joining the midpoints of the sides of the unit square. Since each side of the square has length 1, the sides of the rhombus also have length 1.
The diagonals of a rhombus bisect each other at right angles. Since the sides of the rhombus have length 1, the diagonals are also of length 1.
Now, let's draw the rhombus:
```
A
D-----C
| |
| O |
| |
B-----E
F
```
Let A, B, C, and D be the vertices of the rhombus, and let E be the midpoint of AB and F be the midpoint of BC. The circle with the largest diameter that can be inscribed within the rhombus is tangent to the sides of the rhombus at the midpoints E and F.
Let O be the center of the circle. Since the circle is tangent to the sides of the rhombus at E and F, the radii of the circle are perpendicular to the sides EF and CF.
Since EF is the diagonal of a square with side length 1, EF has length sqrt(2).
Since the diagonals of a rhombus bisect each other at right angles, the length of CF is half the length of the diagonal AC. Since AC has length 1, CF has length 1/2.
Using the Pythagorean theorem, we can find the length of OF:
OF^2 = OC^2 - CF^2
OF^2 = 1/2^2 - 1/2^2
OF^2 = 1/4 - 1/4
OF^2 = 0
This means that OF has length 0, so the circle with the largest diameter that can be inscribed within the rhombus is a point circle. The diameter of this circle is 0.
Therefore, the diameter of the largest circle that can be inscribed within the rhombus is 0.
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Community Answer
A rhombus is formed by joining the midpoints of the sides of a unit sq...
Let ABCD be the square of unit length and P, Q, R, S be midpoints of AB, BC, CD, DA respectively
Let O be centre of circle
OR = OQ = 1/2 units
SP = PQ = RQ = PS (rhombus)
Area of ΔORQ = 1/2OROQ
= 1/2O x RQ
Diameter = 2r = 1/√2
Hence, the correct option is (A).
Let O be centre of circle
OR = OQ = 1/2 units
SP = PQ = RQ = PS (rhombus)
Area of ΔORQ = 1/2OROQ
= 1/2O x RQ
Diameter = 2r = 1/√2
Hence, the correct option is (A).
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A rhombus is formed by joining the midpoints of the sides of a unit square.Q. What is the diameter of the largest circle that can be inscribed within the rhombus?a)1/√2b)1/2√2c)√2d)2√2Correct answer is option 'A'. Can you explain this answer?
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