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Problem-solving Pack
MathematicsYear 9 · 9.5 Coordinate Geometry
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
Line through two points. The line AB passes through and .
(a) Find the gradient as a fraction in simplest form.
(b) Find the equation in form.
(c) State the equation of a line parallel to AB through the origin.
(a) Find the gradient as a fraction in simplest form.
(b) Find the equation in form.
(c) State the equation of a line parallel to AB through the origin.
Working space
2Problem 2 of 12
Dividing a segment. Point C divides AB in ratio 2:1, where , .
Find C.
Find C.
Working space
3Problem 3 of 12
Distance and midpoint together. Two villages at and on a map (km).
(a) Find the straight-line distance.
(b) Find the midpoint.
(c) A new road perpendicular to AB through the midpoint. Find its equation.
(a) Find the straight-line distance.
(b) Find the midpoint.
(c) A new road perpendicular to AB through the midpoint. Find its equation.
Working space
4Problem 4 of 12
Mid-segment investigation. A quadrilateral with vertices A(0, 0), B(6, 0), C(8, 4), D(2, 6).
(a) Find the midpoints P, Q, R, S of sides AB, BC, CD, DA.
(b) Show PQRS is a parallelogram by comparing gradients.
(a) Find the midpoints P, Q, R, S of sides AB, BC, CD, DA.
(b) Show PQRS is a parallelogram by comparing gradients.
Working space
5Problem 5 of 12
Perpendicular bisector. Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6).
Working space
6Problem 6 of 12
Triangle on coordinate plane. Find the perimeter of the triangle with vertices A(0, 0), B(4, 3), C(8, 0).
Working space
7Problem 7 of 12
Distance and perpendicular distance. A point and a line .
(a) Find the equation of the line through P perpendicular to the given line.
(b) Find the foot of the perpendicular (i.e. the point on the line closest to P).
(c) Find the perpendicular distance from P to the line.
(a) Find the equation of the line through P perpendicular to the given line.
(b) Find the foot of the perpendicular (i.e. the point on the line closest to P).
(c) Find the perpendicular distance from P to the line.
Working space
8Problem 8 of 12
Investigating parallel/perpendicular. Lines and .
(a) For what value of are and parallel?
(b) For what value(s) of are they perpendicular?
(a) For what value of are and parallel?
(b) For what value(s) of are they perpendicular?
Working space
9Problem 9 of 12
Triangle bounded by three lines. Sketch the triangle bounded by , , .
(a) Find the three vertices.
(b) Find the area.
(a) Find the three vertices.
(b) Find the area.
Working space
10Problem 10 of 12
Identifying a parallelogram. Show that the points A(1, 1), B(4, 1), C(6, 4), D(3, 4) form a parallelogram.
Working space
11Problem 11 of 12
Centroid of a triangle. A triangle has vertices A(0, 0), B(6, 0), C(3, 6).
(a) Find the centroid (average of vertices).
(b) Find the medians and verify they all pass through the centroid.
(a) Find the centroid (average of vertices).
(b) Find the medians and verify they all pass through the centroid.
Working space
12Problem 12 of 12
Reflection of a line. The line is reflected in the -axis.
(a) Find the equation of the image.
(b) Find the equation when reflected in the -axis.
(a) Find the equation of the image.
(b) Find the equation when reflected in the -axis.
Working space