Is it true to say that area of a square inscribed in a circle of diameter p cm is p2cm2? Why?
False
In the figure we see that the diameter of circle is equal to diagonal of square
Hence, diagonal of square = p cm
Let side of square = a cm Using Pythagoras theorem we get
Area of square = side × side
Here we found that area of square is not equal to p2cm2.
Hence the given statement is False
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Areas of two circles are equal. Is it necessary that their circumferences are equal? Why?
True
Solution
Use area of circle
Let two circles having radius r1 and r2
Here it is given that their areas are equal
We know that circumference of circle =
Circumference of circle with radius r1 = 2πr1
Circumference of circle with radius r2 = 2πr2 …..(1)
Put r2 = r1 in (1) we get
2πr1 = 2πr2
Hence the circumference of given circle are also equal because two circles with equal radii will also have equal circumference.
Therefore, the given statement is True.
View Full Answer(1)Circumferences of two circles are equal. Is it necessary that their areas be equal? Why?
True
Solution
Use circumference of circle =
Let two circles having radius r1 and r2
Here it is given that their circumferences are equal
We know that area of circle =
Area of circle with radius r1 = πr12
Area of circle with radius r2 = πr22 ……..(1)
Put r2 = r1 in (1) we get
πr22 = πr12
Hence the area of given circles are also equal because two circles with equal radii will also have equal areas.
Hence the given statement is True.
View Full Answer(1)Is the area of the largest circle that can be drawn inside a rectangle of length acm and breadth b cm (a >b) is b2 cm2?Why?
False
Solution
Diameter of circle = b
Radius =
Area =
Here we found that the area of the largest circle is not equal πb2cm2.
Hence the given statement is False
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The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal? Why?
True
Solution
Let the radius of first circle is r1 and of other is r2
Let the arc length of both circles are same.
Let the arc length is a.
length of arc (a)=
Area of sector of first circle =
(because area of sector = )
Area of sector of other circle =
Here we found that both areas are equal in the case of when r1 = r2
Hence the area of two sectors of two different circles would be equal only in case of both the circles have equal radii and equal corresponding arc length.
Hence it is necessary that their corresponding arcs lengths are equal.
View Full Answer(1)The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why?
False
Solution.
Let the radius of first circle is r1 and of other is r2
The length of arcs of both circles is same.
Let the arc length = a.
length of arc (a)=
Area of sector of first circle = (because area of sector = )
Area of sector of second circle =
Here we found that the area of sector is depending on radius of circles.
When the circle is same then radius is also same then the given statement is true.
But in case of different circles then the radius is also different
Hence the given statement is False.
View Full Answer(1)If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement false? Why?
True
Solution
Formula of length of arc=
Let Radius of first circle = r
Length of arc = ….. (1) { is the angle of first circle}
Radius of second circle = 2r
Length of arc=
= …..(2) { is the angle of second circle}
According to question
No, this statement is True
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The numerical value of the area of a circle is greater than the numerical value of its circumference. Is this statement true? Why?
False
Solution
Area of circle =
Circumference of circle =
Case 1:
Let r = 1
Area of circle ==
Circumference of circle = =
Case 2:
Let r = 3
Area of circle = πr2 = π(3)2 = 9π
Circumference of circle = 2πr = 2π(3) = 6π
Conclusion:- In case (1) we found that the area is less than the circumference but in case (2) we found that the area is greater than the circumference.
So, from conclusion we observe that it depend on the value of radius of the circle.
Hence the given statement false.
View Full Answer(1)In covering a distance s metres, a circular wheel of radius r m makes revolution. Is this statement true? Why?
[True]
Solution
Circumference of circle =
Radius of circular wheel = r m
Circumference of wheel =
Distance covered in One revolution= circumference of wheel =
In covering a distance of s number of revolution required =
Hence the given statement is True
View Full Answer(1)Is it true that the distance travelled by a circular wheel of diameter d cm in one revolution is 2d cm? Why?
False
Solution
Circumference of circle =2r
Diameter = d
Radius =
Circumference =2r
=
Here we found that the distance travelled by a circular wheel of diameter d cm in one revolution is πd which is not equal to 2πd.
Hence the given statement is False.
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