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Geometry Toolkit (DP IB Analysis & Approaches (AA))

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  • FrontCoordinate Geometry

    State the expression for the midpoint of two points open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses.

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  • State the expression for the midpoint of two points open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses.

    The expression for the midpoint of two points open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses is open parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction comma fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction close parentheses.

    This is given in your exam formula booklet.

  • What is the formula for calculating the distance between two points open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses?

    The distance between two points is calculated using the formula, d equals square root of open parentheses open parentheses x subscript 1 minus x subscript 2 close parentheses squared plus open parentheses y subscript 1 minus y subscript 2 close parentheses squared close parentheses end root

    Where:

    • d is the distance between two points

    • open parentheses x subscript 1 comma space y subscript 1 close parentheses is the set of coordinates for a known point

    • open parentheses x subscript 2 comma space y subscript 2 close parentheses is the set of coordinates for another known point

    This is given in your exam formula booklet.

  • What is the formula for finding the gradient of a line between two points open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses?

    The gradient can be found using the formula m equals fraction numerator open parentheses y subscript 2 minus y subscript 1 close parentheses over denominator open parentheses x subscript 2 minus x subscript 1 close parentheses end fraction

    Where:

    • m is the gradient of the line

    • open parentheses x subscript 1 comma space y subscript 1 close parentheses is the set of coordinates for a known point

    • open parentheses x subscript 2 comma space y subscript 2 close parentheses is the set of coordinates for another known point

    This is given in your exam formula booklet.

  • What does the notation [AB] represent in coordinate geometry?

    The notation [AB] represents the line segment between points A and B.

  • What are radians?

    Radians are a way of measuring of angles as an alternative to degrees.

    1 radian is the angle in a sector with radius 1 and arc length 1.

    Diagram of a circle with center O, radius of 1. PQ is a minor arc subtending an angle of 1 radian at O. Labels show "1 rad ≈ 57.3°" and "Radius 1".

  • What is the symbol for radians?

    Radians are indicated withblank to the power of c but it is more typical to see rad. Sometimes, if the angle includes pi, then no symbol is used as the use of radians is implied.

    Degrees will always be indicated with º.

  • True or False?

    2 pi space rad equals 180 º.

    False.

    2 pi space rad equals 360 º.

  • How do you convert an angle from radians to degrees?

    To convert from radians to degrees, multiply by 180 over pi .

  • How do you convert an angle from degrees to radians?

    To convert from degrees to radians, multiply by pi over 180 .

  • True or False?

    In exams, you should always use degrees unless otherwise indicated.

    False.

    In exams, you should use radians unless otherwise indicated.

  • Define the term arc.

    An arc is a part of the circumference of a circle.

  • Define the term radius.

    The radius is the distance from the centre of a circle to the circumference.

    Radius also refers to a line segment from the centre of a circle to the circumference.

  • Define the term sector.

    A sector is a part of a circle enclosed by two radii and an arc.

  • True or False?

    A minor arc has an angle at the centre less than 180° open parentheses pi space rad close parentheses.

    True.

    A minor arc has an angle at the centre less than 180° open parentheses pi space rad close parentheses.

  • State the equation for the length of an arc when working with degrees.

    The equation for the length of an arc, when working with degrees, is l equals theta over 360 cross times 2 pi r

    Where:

    • l is the length of the arc

    • theta is the angle of the sector in degrees

    • r is the radius of the sector

    This formula is not in the exam formula booklet.

  • State the equation for the length of an arc when working with radians.

    The equation for the length of an arc, when working with radians, is l equals r theta

    Where:

    • l is the length of the arc

    • theta is the angle of the sector in radians

    • r is the radius of the sector

    This formula is in the exam formula booklet.

  • True or False?

    The perimeter of a sector is just the arc length.

    False.

    The perimeter of a sector is the arc length plus two radii.

  • State the equation for the area of a sector, when working with degrees.

    The equation for the area of a sector, when working with degrees, is A equals theta over 360 cross times pi r squared

    Where:

    • A is the area of the sector

    • theta is the angle of the sector in degrees

    • r is the radius of the sector

    This formula is not in the exam formula booklet.

  • State the equation for the area of a sector, when working with radians.

    The equation for the area of a sector, when working with radians, is A equals 1 half r squared theta

    Where:

    • A is the area of the sector

    • theta is the angle of the sector in radians

    • r is the radius of the sector

    This formula is in the exam formula booklet.

  • What is the fraction used to calculate sector area or arc length, when working with degrees?

    The fraction used to calculate sector area or arc length is the angle at the centre divided by 360°, theta over 360.

  • True or False?

    The area of a major sector is always larger than the area of a minor sector in the same circle.

    True.

    The area of a major sector is always larger than the area of a minor sector in the same circle.

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    1. Topic 1

      Number & Algebra Toolkit

    2. Topic 2

      Exponentials & Logs

    3. Topic 3

      Sequences & Series

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      Simple Proof & Reasoning

    5. Topic 5

      Proof by Induction & Contradiction

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      Binomial Theorem

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    1. Topic 1

      Geometry Toolkit

    2. Topic 2

      Geometry of 3D Shapes

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      Trigonometry

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      Statistics Toolkit

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      Normal Distribution

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      Continuous Random Variables

    1. Topic 1

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      Techniques & Applications of Differentiation

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      Maclaurin Series

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      Limits using l'Hôpital's Rule & Maclaurin Series