IGCSE · Mathematics 0580 · Coordinate geometry

Length & midpoint: IGCSE Mathematics questions with answers

Free Length & midpoint questions for IGCSE Mathematics (0580), Extended tier (Supplement content), part of Coordinate geometry. Each question has the answer and the marking points an examiner would look for.

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Length & midpoint practice questions

Question 1Structured · 9 marksA* challenge

The circle x2 + y2 = 50 and the line y = 2x − 5 intersect at P and Q.

(a)

Find the coordinates of P and Q.

[4]
(b)

Find the exact length of PQ, giving your answer in the form k√2 or k√n.

[2]
(c)

Find the equation of the tangent to the circle at P.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(5, 5) and (−1, −7)
substitute the line into the circle M1
three-term quadratic = 0 A1
solve M1
both points A1
4
(b)6√5
√(62 + 122) M1
6√5 A1
2
(c)y = −x + 10
radius gradient 1 M1
perpendicular gradient −1 M1
equation through P A1
3
Question 2Structured · 5 marks

The vertices of triangle ABC are A(1, 1), B(7, 9) and C(13, 1).

(a)

Show that the triangle is isosceles.

[3]
(b)

Find the area of the triangle.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)AB = BC
AB = √(62 + 82) M1
BC = √(62 + 82) M1
AB = BC so isosceles A1
3
(b)48
½ × 12 × 8 M1
48 A1
2
Question 3Structured · 4 marks

A is the point (−4, −5) and B is the point (2, 2).

(a)

Find the coordinates of the midpoint of AB.

[2]
(b)

Find the length of AB.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(−1, −1.5)
One coordinate correct B1
Both correct B1
2
(b)9.22
√(62 + 72) M1
9.22 A1
2
Question 4Structured · 9 marksA* challenge

The circle x2 + y2 = 65 and the line y = x + 3 intersect at P and Q.

(a)

Find the coordinates of P and Q.

[4]
(b)

Find the exact length of PQ, giving your answer in the form k√2 or k√n.

[2]
(c)

Find the equation of the tangent to the circle at P.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(4, 7) and (−7, −4)
substitute the line into the circle M1
three-term quadratic = 0 A1
solve M1
both points A1
4
(b)11√2
√(112 + 112) M1
11√2 A1
2
(c)y = −47x + 657
radius gradient 74 M1
perpendicular gradient −47 M1
equation through P A1
3
Question 5Structured · 5 marks

The vertices of triangle ABC are A(−2, −1), B(1, 3) and C(4, −1).

(a)

Show that the triangle is isosceles.

[3]
(b)

Find the area of the triangle.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)AB = BC
AB = √(32 + 42) M1
BC = √(32 + 42) M1
AB = BC so isosceles A1
3
(b)12
½ × 6 × 4 M1
12 A1
2
Question 6Structured · 4 marks

A is the point (−3, 3) and B is the point (−7, 9).

(a)

Find the coordinates of the midpoint of AB.

[2]
(b)

Find the length of AB.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(−5, 6)
One coordinate correct B1
Both correct B1
2
(b)7.21
√(42 + 62) M1
7.21 A1
2

More Coordinate geometry questions

Gradient & equation of a lineParallel linesPerpendicular lines

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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in