Circle properties

The task is to choose the statement that gives the correct geometric meaning of a sector of a circle. This is a key idea in Right Triangles and Circle Geometry.

A sector is the part of a circle’s interior formed by two radii and the arc between them.

It looks like a slice taken from the center outward, similar to a wedge shape.

This choice describes a straight line joining two points on the circle. That object is called a chord, not a region of area.

This option is the area between a chord and its matching arc. That region is known as a segment of a circle, not a sector.

The phrase refers to the entire length around the circle’s edge. That measurement is the circumference, not a portion of the circle’s interior.

A sector must be formed by two radii and the arc connecting them, so the correct choice is the one that states exactly that.

To determine the definition of a sector, we will check each description and match it to the correct circle term. By crossing out mismatches, the correct definition will remain.

Option A describes a line segment connecting two points on the circumference, which is a chord. Since a sector is an area (a region), not just a line, this cannot be correct.

Option C is the region between a chord and its arc, which defines a segment. A sector specifically involves radii from the center, so this description does not fit.

Option D refers to the total distance around the circle, called the circumference. This is a perimeter measure, not a part of the circle’s interior, so it is not a sector.

After removing the other options, the remaining description is the region enclosed by two radii and an arc. That is exactly what a sector is (like a single slice from the center).

Start by imagining the shape created when you draw lines from the circle’s center to each endpoint of the chord. Those two lines form two sides of a triangle, and the chord is the third side.

The triangle’s sides are:

1. Radius from center to one endpoint: $10\text{ cm}$
2. Radius from center to the other endpoint: $10\text{ cm}$
3. The chord itself: $10\text{ cm}$ So every side length matches.

A triangle with three equal sides is an equilateral triangle. One important fact about equilateral triangles is that all their interior angles are the same size.

Any triangle has angles adding to $180^{\circ }$. If all three angles are equal, then each is one third of $180^{\circ }$:

$\theta =\frac{180^{\circ }}{3}=60^{\circ }$

Compare the options with the result $\theta =60^{\circ }$:

• Option A: If $\theta$ were $90^{\circ }$, the triangle would be right-isosceles. Using Pythagoras, the chord would be $\sqrt{10^2+10^2}\approx 14.1\text{ cm}$, not $10\text{ cm}$.
• Option B and C: These angles don’t make the chord equal to the radius.

Whenever a chord is the same length as the radius, the two radii and the chord form an equilateral triangle, so the central angle is $60^{\circ }$.

Think of the triangle formed by joining the center to the chord’s endpoints. The two equal sides are radii, and the third side is the chord.

Given:

• Side $a$ (radius) = $10$ cm
• Side $b$ (radius) = $10$ cm
• Side $c$ (chord) = $10$ cm

The angle opposite the chord is the central angle $\theta$.

Apply the Law of Cosines:

$c^2=a^2+b^2-2ab\cos (C)$

Here, $c$ is the chord and $a$ and $b$ are the radii, so:

$10^2=10^2+10^2-2(10)(10)\cos (\theta )$

Simplify and solve for $\cos (\theta )$:

$\begin{array}{rl}100& =100+100-200\cos (\theta )\\ 100& =200-200\cos (\theta )\\ -100& =-200\cos (\theta )\\ \cos (\theta )& =\frac{-100}{-200}=0.5\end{array}$

Then compute the angle:

$\theta =\arccos (0.5)$

Since the work gives $\theta =60^{\circ }$, check the others:

• Option A is $90^{\circ }$, but $\cos (90^{\circ })=0$.
• Option B is $45^{\circ }$, but $\cos (45^{\circ })\approx 0.707$.
• Option C is $30^{\circ }$, but $\cos (30^{\circ })\approx 0.866$.

Because the sides are $10$ cm, $10$ cm, and $10$ cm, the triangle is equilateral, so each internal angle is $60^{\circ }$. That agrees with the cosine rule result.

Join the center of the circle to both endpoints of the chord. This makes an isosceles triangle with two radii of $10\text{ cm}$ and a base (the chord) of $10\text{ cm}$. This connects circle facts with right-triangle trigonometry.

Draw a perpendicular from the center down to the chord. This creates two identical right triangles and also halves:

• the central angle to $\frac{\theta }{2}$
• the chord to $\frac{10}{2}=5\text{ cm}$

The hypotenuse of each right triangle is still the radius, $10\text{ cm}$.

Use the sine relationship:

$\sin (\text{angle})=\frac{\text{opposite}}{\text{hypotenuse}}$

So in this right triangle:

$\sin \left(\frac{\theta }{2}\right)=\frac{5}{10}=0.5$

Find the half-angle first, then double it:

$\begin{array}{rl}\frac{\theta }{2}& =\arcsin (0.5)\\ \frac{\theta }{2}& =30^{\circ }\\ \theta & =60^{\circ }\end{array}$

The other choices don’t match the needed right-triangle ratios:

• Option A: With $90^{\circ }$, the chord would be $10\sqrt{2}\approx 14.1\text{ cm}$.
• Option B: With $45^{\circ }$, you’d have $\sin (22.5^{\circ })=\frac{c}{20}$, so the chord would be much smaller than $10\text{ cm}$.
• Option C: With $30^{\circ }$, you’d have $\sin (15^{\circ })=\frac{c}{20}$, which does not give a $10\text{ cm}$ chord.

The calculations show $\theta =60^{\circ }$. Geometrically, that also fits because the two radii and the chord are all $10\text{ cm}$, forming an equilateral triangle with $60^{\circ }$ angles.