c1 Circles question? OCR Mei

liverpool2044

11 The points A (−1, 6), B(1, 0) and C(13, 4) are joined by straight lines
(i) Prove that the lines AB and BC are perpendicular. [3]

(ii) Find the area of triangle ABC. [3]

(iii) A circle passes through the points A, B and C. Justify the statement that AC is a diameter of thiscircle. Find the equation of this circle. [6]

(iv) Find the coordinates of the point on this circle that is furthest from B.

Struggling to do part iii, can someone give me a step by step solution, that would be great, thanks.

Reply 1

TeeEm

19

Original post by liverpool2044

11 The points A (−1, 6), B(1, 0) and C(13, 4) are joined by straight lines
(i) Prove that the lines AB and BC are perpendicular. [3]

(ii) Find the area of triangle ABC. [3]

(iii) A circle passes through the points A, B and C. Justify the statement that AC is a diameter of thiscircle. Find the equation of this circle. [6]

(iv) Find the coordinates of the point on this circle that is furthest from B.

Struggling to do part iii, can someone give me a step by step solution, that would be great, thanks.

think about one famous circle theorem

Reply 2

liverpool2044

OP

10

I understand that its going to be based around finding the hypotenuse,
so my working out so far is:

AC is diameter due to being perpendicular to point B line
midpoint (6,5)
Radius = 0.5 x Square root of 14^2 +2^2 = 0.5root200

not sure where to go from here

OP

??

Original post by liverpool2044

I understand that its going to be based around finding the hypotenuse,
so my working out so far is:

AC is diameter due to being perpendicular to point B line
midpoint (6,5)
Radius = 0.5 x Square root of 14^2 +2^2 = 0.5root200

not sure where to go from here

It doesn't seem like you've considered a circle theorem as TeeEm suggested. Think back to a GCSE circle theorem that involves the diameter of a circle.

The question says 'justify' so you don't really need working for this part.

If you haven't already drawn a diagram, then I recommend drawing one and it may become clearer.

(edited 8 years ago)

Reply 5

liverpool2044

OP

10

where lines join up at diameter = 90 degrees

Reply 6

Notnek

21

Original post by liverpool2044

where lines join up at diameter = 90 degrees

Yes, if those lines meet at the circumference.

The famous way of quoting the theorem is as follows : "Angles in a semicircle are 90 degrees".

Or better in my opinion : "The diameter of a circle always subtends a right-angle at the circumference of the circle"

You would have to check the mark scheme to see what is acceptable.

(edited 8 years ago)

Reply 7

liverpool2044

OP

10

Thank you, what else do i need to do to find the radius using the calculation ive done?

Reply 8

Notnek

21

Original post by liverpool2044

Thank you, what else do i need to do to find the radius using the calculation ive done?

If you know that A and C are endpoints of the diameter, then the diameter length is the length of AC and the radius is half of this length.

EDIT: You've found the centre (6,5), so the radius is simply the distance between (6,5) and one of A or C.