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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 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 A(−2,1)A(-2, 1) and B(7,4)B(7, 4).

(a) Find the gradient as a fraction in simplest form.
(b) Find the equation in y=mx+cy = mx + c 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 A(−2,1)A(-2, 1), B(7,4)B(7, 4).

Find C.

Working space

3Problem 3 of 12
Distance and midpoint together. Two villages at A(2,5)A(2, 5) and B(10,11)B(10, 11) 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.

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.

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 P(5,1)P(5, 1) and a line y=2x−1y = 2x - 1.

(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 ℓ1:y=mx+2\ell_1: y = mx + 2 and ℓ2:y=(3−m)x+5\ell_2: y = (3 - m)x + 5.

(a) For what value of mm are ℓ1\ell_1 and ℓ2\ell_2 parallel?
(b) For what value(s) of mm are they perpendicular?

Working space

9Problem 9 of 12
Triangle bounded by three lines. Sketch the triangle bounded by y=x+1y = x + 1, y=−x+5y = -x + 5, y=0y = 0.

(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.

Working space

12Problem 12 of 12
Reflection of a line. The line y=2x+1y = 2x + 1 is reflected in the xx-axis.

(a) Find the equation of the image.
(b) Find the equation when reflected in the yy-axis.

Working space