Podcast
Questions and Answers
What is the relationship between the diameter ($d$) and the radius ($r$) of a circle?
What is the relationship between the diameter ($d$) and the radius ($r$) of a circle?
- $d = \pi r$
- $d = \frac{r}{2}$
- $d = r^2$
- $d = 2r$ (correct)
A line intersects a circle at two distinct points. What term is used to describe this line?
A line intersects a circle at two distinct points. What term is used to describe this line?
- Tangent
- Secant (correct)
- Chord
- Radius
According to the theorem regarding the perpendicular from the center to a chord, what does this perpendicular do to the chord?
According to the theorem regarding the perpendicular from the center to a chord, what does this perpendicular do to the chord?
- It is tangent to the chord.
- It extends the chord.
- It is parallel to the chord.
- It bisects the chord. (correct)
If an arc $AB$ on a circle subtends an angle of $60°$ at the center $O$, what is the angle it subtends at a point $C$ on the remaining part of the circle?
If an arc $AB$ on a circle subtends an angle of $60°$ at the center $O$, what is the angle it subtends at a point $C$ on the remaining part of the circle?
In a circle with center $O$, $PT$ is a tangent to the circle at point $P$. What is the measure of angle $∠OPT$?
In a circle with center $O$, $PT$ is a tangent to the circle at point $P$. What is the measure of angle $∠OPT$?
What is the formula to calculate the length of an arc, given the radius $r$ and the angle $\theta$ (in degrees) subtended by the arc at the center?
What is the formula to calculate the length of an arc, given the radius $r$ and the angle $\theta$ (in degrees) subtended by the arc at the center?
Two tangents are drawn to a circle from an external point. If the length of one tangent is 8 cm, what is the length of the other tangent from the same point?
Two tangents are drawn to a circle from an external point. If the length of one tangent is 8 cm, what is the length of the other tangent from the same point?
What is the formula for calculating the area of a sector of a circle, given the radius $r$ and the angle $\theta$ (in degrees)?
What is the formula for calculating the area of a sector of a circle, given the radius $r$ and the angle $\theta$ (in degrees)?
Flashcards
What is a circle?
What is a circle?
Set of all points equidistant from a center.
What is the center of a circle?
What is the center of a circle?
The fixed point inside the circle.
What is the radius?
What is the radius?
Distance from the center to any point on the circle.
What is the diameter?
What is the diameter?
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What is a chord?
What is a chord?
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What is a tangent?
What is a tangent?
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What is an arc?
What is an arc?
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What is the circumference?
What is the circumference?
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Study Notes
- A circle is a two-dimensional shape.
- It includes all points in a plane equidistant from a center point.
- The radius is the distance from the center to any point on the circle.
Key Terms
- Center (O): The fixed point inside the circle.
- Radius (r): Distance between the center and any point on the circle.
- Diameter (d): Longest chord, passing through the center; d = 2r.
- Chord: A line segment joining two points on the circle.
- Secant: A line intersecting the circle at two points.
- Tangent: A line touching the circle at exactly one point.
- Arc: Part of the circumference of a circle.
- Circumference: Total distance around the circle; C = 2Ï€r.
- Sector: Region bounded by two radii and an arc.
- Segment: Region enclosed by a chord and an arc.
Theorems
- Perpendicular from the Center to a Chord:
- A perpendicular line from the center of a circle to a chord bisects the chord.
- Equal Chords and Their Distances from the Center:
- Chords equidistant from the center are equal in length.
- Angle Subtended by an Arc at the Center:
- The angle at the center is twice the angle at any point on the remaining part of the circle.
- ∠AOB = 2 × ∠ACB, where O is the center.
- Angles in the Same Segment:
- Angles in the same segment of a circle are equal.
- Angle in a Semicircle:
- The angle in a semicircle is always 90°.
- Tangent to a Circle:
- A tangent is perpendicular to the radius at the point of contact.
- OP ⊥ PT, where O is the center and P is the point of contact.
- Length of Tangents from an External Point:
- Tangents from an external point to a circle are equal in length.
Circle Formulas
- Circumference = 2Ï€r
- Area of Circle = πr²
- Length of Arc = (θ/360°) × 2πr
- Area of Sector = (θ/360°) × πr²
- Area of Segment = Area of Sector – Area of Triangle formed
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Description
Explore the fundamental elements of a circle, including its center, radius, and diameter. Key concepts such as chords, secants, and tangents are defined. Important theorems related to chords and their distances from the center of the circle are discussed.
