Plane geometry circles
55 flashcards covering Plane geometry circles for the GMAT Quantitative section.
Plane geometry circles involve the study of round shapes in a flat, two-dimensional plane. A circle is defined as the set of all points equidistant from a central point, known as the center. Key elements include the radius (distance from center to edge), diameter (twice the radius), circumference (the perimeter), and area. You'll also explore properties like chords (straight lines connecting two points on the circle), tangents (lines touching the circle at one point), and arcs (portions of the circumference). Understanding these basics helps build a strong foundation for solving geometric problems in everyday math and beyond.
On the GMAT Quantitative section, circles often appear in problem-solving and data sufficiency questions that require applying formulas and theorems, such as calculating areas, angles, or intersections with lines and other shapes. Common traps include confusing radius with diameter, overlooking right triangles formed by radii and chords, or misinterpreting inscribed versus circumscribed figures. Focus on mastering key formulas like the area (πr²) and circumference (2πr), as well as properties of tangents and central angles, to handle these efficiently under time pressure.
Practice drawing accurate diagrams for every problem.
Terms (55)
- 01
Circle
A circle is the set of all points in a plane that are at a fixed distance from a fixed point called the center.
- 02
Radius
The radius of a circle is the distance from the center to any point on the circle.
- 03
Diameter
The diameter of a circle is a straight line passing through the center and connecting two points on the circle, equal to twice the radius.
- 04
Circumference
The circumference is the perimeter of a circle, calculated using the formula C = 2πr, where r is the radius.
- 05
Area of a circle
The area of a circle is the space enclosed by its circumference, given by the formula A = πr², where r is the radius.
- 06
Chord
A chord is a straight line segment with endpoints on the circle.
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Arc
An arc is a portion of the circumference of a circle, defined by two endpoints on the circle.
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Minor arc
A minor arc is the shorter arc connecting two points on a circle, with a measure less than 180 degrees.
- 09
Major arc
A major arc is the longer arc connecting two points on a circle, with a measure greater than 180 degrees.
- 10
Central angle
A central angle is an angle formed by two radii at the center of the circle, with its measure equal to the arc it subtends.
- 11
Inscribed angle
An inscribed angle is an angle formed by two chords sharing a common endpoint on the circle, with its measure half the arc it subtends.
- 12
Tangent to a circle
A tangent is a straight line that touches the circle at exactly one point, called the point of tangency, and is perpendicular to the radius at that point.
- 13
Secant
A secant is a straight line that intersects the circle at two points.
- 14
Point of tangency
The point of tangency is the single point where a tangent line meets the circle.
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Arc length
Arc length is the distance along the circumference between two points on a circle, calculated as (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius.
- 16
Sector
A sector is a region bounded by two radii and the arc between them, like a slice of pizza.
- 17
Area of a sector
The area of a sector is the portion of the circle's area enclosed by two radii and their arc, calculated as (θ/360) × πr², where θ is the central angle in degrees and r is the radius.
- 18
Segment of a circle
A segment is the region bounded by a chord and the arc it subtends, consisting of the area between the chord and the circumference.
- 19
Area of a segment
The area of a segment is calculated by subtracting the area of the triangular portion from the area of the sector that includes the same central angle.
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Inscribed quadrilateral
An inscribed quadrilateral is a four-sided polygon where all vertices lie on the circle, and the sum of each pair of opposite angles is 180 degrees.
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Cyclic quadrilateral
A cyclic quadrilateral is the same as an inscribed quadrilateral, meaning all its vertices lie on a single circle.
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Equation of a circle
The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r² in the coordinate plane.
- 23
Intersecting chords theorem
The intersecting chords theorem states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal.
- 24
Tangent-secant theorem
The tangent-secant theorem states that if a tangent and a secant are drawn from an external point to a circle, the square of the tangent segment equals the product of the secant segment and its external part.
- 25
Power of a point
Power of a point is a theorem that relates the lengths of segments created by chords, secants, or tangents from a point outside or inside the circle.
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Two tangents from an external point
From an external point, two tangents to a circle are equal in length, and the line from the point to the center bisects the angle between them.
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Length of a tangent from a point
The length of a tangent from an external point to the point of tangency is given by √(d² - r²), where d is the distance from the external point to the center, and r is the radius.
- 28
Concentric circles
Concentric circles are two or more circles that share the same center but have different radii.
- 29
Common chord
A common chord is a line segment that is a chord in two intersecting circles.
- 30
Distance between centers of two circles
The distance between the centers of two circles determines if they are separate, tangent, intersecting, or one inside the other.
- 31
Externally tangent circles
Two circles are externally tangent if they touch at exactly one point and their centers are separated by a distance equal to the sum of their radii.
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Internally tangent circles
Two circles are internally tangent if they touch at exactly one point and their centers are separated by a distance equal to the difference of their radii.
- 33
Shaded region between two circles
A shaded region between two circles, such as an annulus, is the area between their circumferences, calculated as the difference of their areas.
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Intersection points of two circles
The intersection points of two circles can be found by solving their equations simultaneously in the coordinate plane.
- 35
Diameter as hypotenuse
In a circle, if a triangle is inscribed with one side as the diameter, the angle opposite the diameter is a right angle, by the inscribed angle theorem.
- 36
Chord perpendicular to radius
A radius drawn to the midpoint of a chord is perpendicular to the chord, bisecting it into two equal parts.
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Angle subtended by an arc
The measure of an angle subtended by an arc at the center is twice that at any point on the remaining part of the circumference.
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Circumscribed circle
A circumscribed circle passes through all vertices of a polygon, such as a triangle or quadrilateral.
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Inscribed circle
An inscribed circle is tangent to all sides of a polygon, like the incircle of a triangle.
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Strategy for arc length problems
To solve arc length problems, first find the central angle, then use the proportion of the circle's circumference that the arc represents.
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Common trap: Inscribed vs. central angles
A common error is treating an inscribed angle as equal to the central angle; remember, the inscribed angle is half the central angle for the same arc.
- 42
Formula for chord length
The length of a chord is calculated using 2r sin(θ/2), where r is the radius and θ is the central angle in radians subtended by the chord.
- 43
Right triangle in a semicircle
Any triangle inscribed in a semicircle with the diameter as one side is a right triangle, with the right angle at the point on the arc.
- 44
Sector perimeter
The perimeter of a sector includes the two radii and the arc length, useful for problems involving the boundary of the sector.
- 45
Annulus area
The area of an annulus, the region between two concentric circles, is the difference between the areas of the larger and smaller circles.
- 46
Tangent circles strategy
When dealing with tangent circles, use the distance between centers equal to the sum or difference of radii to set up equations for unknowns.
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Overlapping circles area
To find the area of overlap between two intersecting circles, calculate the areas of the circular segments formed by their intersection points.
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Pythagorean theorem in circles
Apply the Pythagorean theorem to triangles formed by radii and chords, such as in right triangles where the hypotenuse is a diameter.
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Example: Arc length calculation
For a circle with radius 5 and a central angle of 72 degrees, the arc length is (72/360) × 2π × 5 = (0.2) × 10π = 2π.
If the radius is 5 and angle is 72°, arc length is about 6.28 units.
- 50
Example: Area of sector
For a circle with radius 10 and a central angle of 90 degrees, the sector area is (90/360) × π × 10² = (0.25) × 100π = 25π.
With radius 10 and 90° angle, sector area is about 78.54 square units.
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Example: Chord length
For a circle with radius 13 and a central angle of 60 degrees, the chord length is 2 × 13 × sin(30°) = 26 × 0.5 = 13.
Radius 13, 60° angle gives a chord of length 13.
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Example: Tangent length
If a point is 13 units from the center of a circle with radius 5, the tangent length is √(13² - 5²) = √(169 - 25) = √144 = 12.
Distance 13, radius 5 yields tangent of 12 units.
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Example: Intersecting chords
If two chords intersect inside a circle with segments 3 and 4 on one chord, and 2 and x on the other, then 3 × 4 = 2 × x, so x = 6.
Segments 3 and 4, and 2 and x, solve for x as 6.
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Example: Inscribed angle
If an arc measures 80 degrees, the inscribed angle subtended by that arc is 40 degrees.
° arc gives a 40° inscribed angle.
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Example: Area of segment
For a circle with radius 10 and a central angle of 60 degrees, subtract the triangle area from the sector area: sector is (60/360)×π×10² = (1/6)×100π, triangle is (1/2)×10×10×sin(60°).
Roughly, segment area is about 22.36 square units.