Circle theorems

The description is of a line that touches the circle’s edge at exactly one point and stays outside the circle rather than passing through it. In geometry, that line is called a tangent line.

A secant goes through a circle and hits the circumference at two points, meaning it enters the interior. Also, “terminus” suggests an end of a segment, not the special touchpoint of a tangent.

A chord is a segment with both endpoints on the circle, typically lying inside the circle. The situation here is a line coming from outside and only touching once, so this term doesn’t match.

Even though the line and the circle meet at a point, “point of intersection” can describe many different crossings or meetings in math. The question asks for the specific formal term used for tangents and circles.

By the standard definition in circle geometry (covered in Topic: Right Triangles and Circle Geometry), the single point where a tangent touches a circle is called the point of tangency.

We need the formal name of the point where a line touches the circle exactly once and stays outside the circle rather than passing through it.

A secant must cut through the circle and meet it at two points. Since the line in the question does not enter the interior, any answer involving a secant cannot be correct.

A chord connects two points on the circumference and lies within the circle. Our line is not a segment inside the circle, so this option does not describe the situation.

“Point of intersection” is a broad term that applies whenever two graphs or shapes meet. The question asks for the special term used specifically for a tangent touching a circle.

After eliminating the other choices, we use the definition: a tangent touches a circle at exactly one point without entering it. That point is the point of tangency.

This idea is central to Topic 7: Right Triangles and Circle Geometry in the MYP syllabus.

Before choosing an answer, we should separate two common circle regions: sectors and segments.

• A sector is the part of a circle formed by two radii and an arc (like a wedge or slice).
• A segment is the part of a circle formed by a chord and an arc.

The region is described as being bounded by the chord. That means the region must be a segment, because sectors require radii as boundaries. So any option that is a sector can be ruled out.

A chord splits a circle into two segments:

1. The segment with the shorter curve is the minor segment (it uses the minor arc).
2. The segment with the longer curve is the major segment (it uses the major arc).

Because the question states the region is bounded by the major arc, it cannot be the minor segment.

By definition, the region enclosed by a chord together with the major arc is called the major segment.

Start by focusing on the description: it is the region bounded by the chord and the major arc. That tells us exactly what kind of boundary we are dealing with.

A sector is formed by two radii plus an arc. Since this region uses a chord (not radii), any option mentioning a sector does not fit and can be removed.

A segment is correct for a chord-and-arc region, but there are two sizes:

1. Minor segment: bounded by the smaller (minor) arc.
2. Major segment: bounded by the larger (major) arc.

Because the problem specifies the major arc, the minor segment must be eliminated.

After eliminating the sector choices and the minor segment, the remaining option that matches a chord with the major arc is Major segment.

We need the rule that links the angle formed at the center of a circle by a given arc with the angle formed at the circumference by that same arc. This is a key fact from Circle Geometry.

In the IB MYP topic Right Triangles and Circle Geometry, the Central Angle Theorem (also called the Angle at the Center Theorem) tells us: the angle at the center formed by two radii is exactly twice the angle at the circumference formed by chords that intercept the same arc endpoints.

Option A says the central angle is half the inscribed angle. The theorem goes the other way: the angle on the circumference is half the central angle, so this choice is incorrect.

Option B claims the central angle is four times the inscribed angle. That multiplier does not match any standard circle theorem for these two angles, so it cannot be right.

Option C says the angles are the same. Angles standing on the same arc at the circumference can be equal (Angles in the Same Segment Theorem), but the central angle is always larger than the corresponding angle at the edge.

By the Central Angle Theorem, the angle at the center is exactly twice the angle at the circumference.

Mathematically, if the angle at the circumference is $\theta$, the angle at the center is $2\theta$.

The statement compares two angles made by the same arc:

1. The angle subtended by an arc at the center (central angle).
2. The angle subtended by the same arc at the circumference (inscribed angle).

To check the relationship, we can test a familiar example: an arc created by a diameter.

Take the arc that is exactly half the circle (a semicircle). This arc is determined by the endpoints of a diameter.

Choosing a diameter makes the angle measures straightforward without needing extra calculations.

A diameter forms a straight line through the center, so the angle at the center is a straight angle of $180^{\circ }$.

$\text{Central Angle}=180^{\circ }$

By Thales's Theorem, any angle subtended by a diameter at the circumference is a right angle.

$\text{Angle at Circumference}=90^{\circ }$

Now compare:

• Central Angle = $180^{\circ }$
• Circumference Angle = $90^{\circ }$

Since $180^{\circ }=2\times 90^{\circ }$, the central angle is twice the circumference angle.

Check the other options against this:

• If it were half, the central angle would be $45^{\circ }$.
• If it were four times, the central angle would be $360^{\circ }$.
• If it were the same, the central angle would be $90^{\circ }$.

This test case shows the central angle ($180^{\circ }$) is exactly twice the angle at the circumference ($90^{\circ }$). Therefore, the Central Angle Theorem’s relationship is confirmed.

Imagine a circle and choose an arc.

1. Mark the center ($O$) and draw radii to the arc’s endpoints to form the central angle.
2. Choose a point on the circumference elsewhere and draw chords to the same endpoints to form the inscribed angle.

This setup comes from the Right Triangles and Circle Geometry part of the MYP course.

The angle at the center opens out more because its vertex is at the middle of the circle. The angle on the circumference, with its vertex on the edge, appears narrower for the same arc.

So the central angle must be bigger than the inscribed angle, which helps eliminate choices.

Since the central angle is clearly larger, it cannot be half of the inscribed angle, and it cannot be the same as the inscribed angle.

That leaves twice and four times. Circle theorems typically involve simple ratios like $2$.

Thinking about a diameter example: the central angle is $180^{\circ }$ while the circumference angle is $90^{\circ }$. Since $180$ is exactly double $90$, twice makes sense and four times is too large.

By the Central Angle Theorem (Angle at the Center Theorem), the angle at the center is exactly twice the angle at the circumference.