Chapter 6 Circles

Question 1

how do you find the midpoint of a line segment?

Answer 1

(x1+x2/2 , y1+y2/2)

Question 2

what is a perpendicular bisector?

Answer 2

the perpendicular bisector of a line segment AB is the straight line that is perpendicular to AB and passes through the midpoint of AB

Question 3

what is the relationship between the gradient of a line and the gradient of a perpendicular line?

Answer 3

if the original gradient is m then the gradient of a line perpendicular to the original is -1/m

Question 4

what is the equation of a circle with centre (0, 0) and radius r?

Answer 4

x^2 + y^2 = r^2

Question 5

what is the equation of a circle with centre (a, b) and radius r?

Answer 5

(x - a)^2 + (y - b)^2 = r^2

Question 6

what is the alternate form of the equation of a circle with centre (a, b) and radius r? (brackets multiplied)

Answer 6

x^2 + y^2 - 2ax - 2by + b^2 + a^2 - r^2 = 0

Question 7

how many times can a straight line intersect a circle?

Answer 7

• a straight line can intersect a circle once, by just touching the circle, or twice
• not all straight lines will intersect a given circle

Question 8

how is a tangent to a circle related to the radius of the circle?

Answer 8

a tangent to a circle is perpendicular to the radius of the circle at the point of intersection

Question 9

how is a chord of a circle related to the centre of the circle?

Answer 9

the perpendiuclar bisector of the chord will go through the centre of a circle

Question 10

what is a circumcircle?

Answer 10

a unqiue circle drawn through the 3 verticies of any triangle

Question 11

what is the circumcentre of a triangle?

Answer 11

the centre of the triangles circumcircle

Question 12

what is the diameter of a circumcircle for a right angle triangle?

Answer 12

the hypotenuse of the triangle is equal to the diameter of the circumcircle so:
- if PRQ = 90, then R lies on the circle with diameter PQ
- the angle in a semicircle is always a right angle

Question 13

how do you find the radius of a circle with centre (0, 0) and with (a, b) lying on the circle?

Answer 13

a^2 + b^2 = r^2

Question 14

how do you find the centre of a circle given any 3 points (a, b, c)?

Answer 14

• find the equations of the perpendicular bisectors of two different chords (e.g. AB, AC or BC)
• find the coordinates of intersection of the perpendicular bisectors