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Questions and Answers
The sum of the interior angles of a quadrilateral is always 540 degrees.
False
A rhombus is defined as an equilateral trapezoid.
False
A rectangle and a square both have all angles equal to 90 degrees.
True
The radius of a circle is always longer than the diameter.
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In a kite, two pairs of unequal adjacent sides are congruent.
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A tangent line intersects a circle at exactly two points.
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The length of the circumference of a circle is independent of its radius.
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A trapezoid can be classified as right, isosceles, or scalene.
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A diagonal in a quadrilateral connects two adjacent vertices.
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In a parallelogram, adjacent angles are always supplementary and opposite angles are equal.
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A secant line intersects a circle at three points.
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The diameter of a circle is equal to the radius multiplied by 2.
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The interior of a circle includes points whose distances from the center are greater than the radius.
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All circles are similar to each other, regardless of their sizes.
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The central angle and the inscribed angle of a circle are equal.
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Every tangent line of a circle is parallel to the radius at the point of tangency.
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The distance between two tangent circles is always equal to the sum of their radii.
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Eccentricity of an ellipse is always greater than 1.
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A chord divides a circle into three segments.
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An equilateral triangle can be inscribed in a circle.
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Study Notes
Quadrilaterals
- A quadrilateral is a four-sided polygon, also known as a tetragon or quadrangle.
- The sum of the interior angles of any quadrilateral is always 360 degrees.
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Parts of a Quadrilateral:
- Sides: Line segments joining two adjacent vertices (corners).
- Interior Angles: Angles formed between two adjacent sides.
- Height or Altitude: The distance between two parallel sides of a quadrilateral.
- Base: The bottom side that is perpendicular to the altitude.
- Diagonal: The line segment that joins two non-adjacent vertices.
Classification of Quadrilaterals
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Parallelogram:
- Two pairs of parallel sides.
- Opposite sides and angles are congruent.
- Adjacent/consecutive angles are supplementary.
- Two diagonals bisect each other.
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Rhombus (Diamond):
- All sides are equal ("equilateral parallelogram").
- Opposite angles are equal, adjacent angles are not equal.
- When diagonals bisect each other, the resulting angle is 90 degrees.
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Rectangle:
- All angles are right angles.
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Square:
- A special type of rectangle.
- All sides are equal, and all angles are right angles.
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Trapezoid:
- Only one pair of parallel sides.
- Types: Right, isosceles, or scalene (trapezium).
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Kite:
- Two pairs of congruent and adjacent sides.
- A special case of a rhombus.
Circles
- A circle is a set of points equidistant from a fixed point called the center.
- It is a closed two-dimensional figure that is perfectly symmetrical.
Parts of a Circle
- Circumference: The length of the outer boundary of the circle. It's the perimeter and represents the "circle-ness" of a circle.
- Radius: The line segment that joins the center of the circle to any point on the circle. It's half the length of the diameter.
- Diameter: Twice the length of the radius. It is the longest chord of the circle.
- Sector: The figure formed by two radii and an included arc.
- Chord: A line segment that joins two points on the circle. It's a segment of a secant line.
- Segment: The region bounded by a chord of a circle and the intercepted arc of the circle.
- Arc: A portion of the circle that contains two endpoints and all the points of the circle between those endpoints.
- Tangent Line: A line that intersects the circle at exactly one point (point of tangency). It touches the circle without crossing it.
- Secant Line: A line that intersects a circle at two points.
Characteristics of Circles
- A chord divides a circle into a major segment (larger part) and a minor segment (smaller part).
- The diameter is the longest chord.
- Circles are congruent if they have equal radii; they are always similar even if their radii are not equal.
- Circles can circumscribe a rectangle, trapezium, triangle, square, and kite.
- The interior of a circle is the set of all points within the boundary whose distances from the center are less than the radius.
- The exterior of a circle is the set of all points outside of the boundary whose distances from the center are greater than the radius.
- The perpendicular distance from the center to the longest chord is zero.
- The perpendicular distance from the center of the circle decreases as the length of the chord increases.
- The central angle is the angle whose vertex lies at the center of the circle, with two sides formed by radii.
- The inscribed angle is the angle whose vertex lies on the circle, with two sides formed by chords of the circle.
Circle Theorems
- Every tangent line of a circle is perpendicular to the radius drawn through the point of tangency.
- Central angles of similar circles have the same ratio as their intercepted arcs.
- You can create an isosceles triangle by joining the ends of a chord to two radii of a circle.
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Lines of Centers of Tangent Circles:
- The line of centers of two tangent circles passes through the point of tangency.
- The distance between the circles (where they are on the same side of the line) is equal to the sum of their radii (r1 + r2).
- The distance between the circles (where they are on opposite sides of the line) is equal to the difference of their radii (r2 - r1).
Miscellaneous Plane Figures
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Star:
- Generally consists of a polygon with triangles on its sides.
- A regular star is referred to as a regular star, German star, or witch star.
- A hexagram is a six-pointed star, known as David's star or Solomon's star.
- Area: Calculate the area of the triangles first, then the area of the remaining polygon. Finally, add the areas together.
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Ellipse:
- A plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.
- Eccentricity is less than 1.
- A circle is a special type of ellipse where the focal points are equal.
- TL;DR: It's oblong.
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Parabolic Section:
- A plane curve that is mirror-symmetrical and approximately U-shaped.
- Eccentricity is equal to 1.
- Spandrel is the almost triangular space between one side of the outer curve of an arch, often used in architecture.
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Description
Test your knowledge on quadrilaterals and their properties with this quiz. Explore the different types of quadrilaterals, including parallelograms and rhombuses, and understand their characteristics and angle relationships.
