Is it true to say that the area of a segment of a circle is less than the area of its corresponding sector? Why?
(B) The statement is false.
$1$. $A$ segment of a circle is the region bounded by a chord and an arc. $A$ sector of a circle is the region bounded by two radii and an arc.
$2$. For a minor segment,the area is indeed less than the area of the corresponding minor sector because the minor segment is a part of the minor sector (the sector includes the triangle formed by the two radii and the chord).
$3$. However,for a major segment,the area is always greater than the area of the corresponding major sector. Therefore,the statement is not universally true.
$1$. $A$ segment of a circle is the region bounded by a chord and an arc. $A$ sector of a circle is the region bounded by two radii and an arc.
$2$. For a minor segment,the area is indeed less than the area of the corresponding minor sector because the minor segment is a part of the minor sector (the sector includes the triangle formed by the two radii and the chord).
$3$. However,for a major segment,the area is always greater than the area of the corresponding major sector. Therefore,the statement is not universally true.
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Similar Questions
Which of the following correctly matches the information given in Part $I$ and Part $II$?
Part $I$ Part $II$ $1.$ Formula to find the length of a minor arc $a.$ $C=2\pi r$ $2.$ Formula to find the area of a minor sector $b.$ $A=\pi r^{2}$ $3.$ Formula to find the area of a circle $c.$ $l=\frac{\pi r \theta}{180}$ $4.$ Formula to find the circumference of a circle $d.$ $A=\frac{\pi r^{2} \theta}{360}$
| Part $I$ | Part $II$ |
| $1.$ Formula to find the length of a minor arc | $a.$ $C=2\pi r$ |
| $2.$ Formula to find the area of a minor sector | $b.$ $A=\pi r^{2}$ |
| $3.$ Formula to find the area of a circle | $c.$ $l=\frac{\pi r \theta}{180}$ |
| $4.$ Formula to find the circumference of a circle | $d.$ $A=\frac{\pi r^{2} \theta}{360}$ |
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