Chapter 6 - Circles

    Cards (29)

    • How can you find the midpoint of a line segment?

      By averaging the x- and y-coordinates of its endpoints
    • What is a line segment?

      A finite part of a straight line with two distinct endpoints
    • What is the perpendicular bisector of a line segment AB?

      It is the straight line that is perpendicular to AB and passes through the midpoint of AB
    • What defines a circle in geometry?

      A circle is the set of points that are equidistant from a fixed point
    • How can you derive equations of circles on a coordinate grid?
      By using Pythagoras' theorem
    • What is the equation of a circle with center (0, 0) and radius r?
      x² + y² = r²
    • What is the general form of the equation of a circle with center (a, b) and radius r?
      (x - a)² + (y - b)² = r²
    • How can a straight line interact with a circle?
      A straight line can intersect a circle once, twice, or not at all
    • What is a tangent to a circle?

      A straight line that intersects the circle at only one point
    • What is the relationship between a tangent and the radius of a circle at the point of intersection?

      A tangent is perpendicular to the radius at the point of intersection
    • What is a chord in a circle?

      A line segment that joins two points on the circumference of a circle
    • What does the perpendicular bisector of a chord do?

      It goes through the center of the circle
    • What is a triangle and its properties related to circles?

      • A triangle consists of three points called vertices.
      • A unique circle can be drawn through the three vertices, called the circumcircle.
      • The center of the circumcircle is the circumcentre, where the perpendicular bisectors of each side intersect.
    • What is the relationship between the hypotenuse of a right-angled triangle and its circumcircle?

      The hypotenuse is a diameter of the circumcircle
    • How can you state the result regarding a right angle in a semicircle?

      If ∠PRQ = 90°, then R lies on the circle with diameter PQ
    • What is the angle in a semicircle property?

      The angle in a semicircle is always a right angle
    • How do you find the center of a circle given three points on the circumference?

      1. Find the equations of the perpendicular bisectors of two different chords.
      2. Find the coordinates of the point of intersection of the perpendicular bisectors.
    • What is the summary of the perpendicular bisector property?

      The perpendicular bisector of a line segment AB is perpendicular to AB and passes through its midpoint
    • What is the summary of the equation of a circle with center (0, 0)?
      The equation is x² + y² = r²
    • What is the summary of the general form of the equation of a circle?

      The equation is x² + y² + 2fx + 2gy + c = 0
    • What is the summary of the tangent property in relation to circles?

      A tangent is perpendicular to the radius at the point of intersection
    • What is the summary of the chord property in relation to circles?

      The perpendicular bisector of a chord goes through the center of the circle
    • What is the summary of the circumcircle property of triangles?

      A unique circle can be drawn through the three vertices of any triangle, called the circumcircle
    • What is the summary of the right-angled triangle property in relation to circumcircles?

      The hypotenuse of a right-angled triangle is a diameter of the circumcircle
    • What is the summary of finding the center of a circle using three points?

      Find the equations of the perpendicular bisectors of two different chords and their intersection
    • What is the formula for the midpoint of a line segment with endpoints (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2)?


      (x1+x22,y1+y22)\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)
    • If the gradient of line segment AB is mm, what is the gradient of its perpendicular bisector?


      1m-\frac{1}{m}
    • How can the equation of a circle also be expressed in the form x2+x² +y2+ y² +2fx+ 2fx +2gy+ 2gy +c= c =0 0?


      This circle has center (-f, -g) and radius f2+g2c\sqrt{f² + g² - c}
    • What is the summary of the midpoint formula?


      The midpoint of a line segment with endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is (x1+x22,y1+y22)\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
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