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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 10 · 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
Triangle in the plane. Points A(1,2)A(1, 2), B(7,2)B(7, 2), and C(4,6)C(4, 6) form a triangle.

(a) Find the lengths ABAB, BCBC, ACAC.
(b) Is the triangle isosceles, equilateral, or scalene?
(c) Find the area of the triangle.

Working space

2Problem 2 of 12
Parallelogram. Three vertices of a parallelogram are A(1,1)A(1, 1), B(5,3)B(5, 3), C(7,7)C(7, 7).

(a) Find the gradient of ABAB and BCBC.
(b) The fourth vertex DD is such that ABCDABCD is a parallelogram (in order). Find DD.
(c) Find the perimeter of the parallelogram in exact form.

Working space

3Problem 3 of 12
Perpendicular bisector. Find the equation of the perpendicular bisector of the line segment joining A(−2,3)A(-2, 3) and B(6,7)B(6, 7).

Working space

4Problem 4 of 12
Right-angled triangle. Show that the triangle with vertices P(−1,−1)P(-1, -1), Q(5,1)Q(5, 1), R(4,4)R(4, 4) is right-angled. At which vertex is the right angle?

Working space

5Problem 5 of 12
Find missing coordinates. A line passes through (2,−1)(2, -1) and has gradient 3.

(a) Find its equation.
(b) Where does this line cross the xx-axis and yy-axis?
(c) Find the area of the triangle formed by this line and the two axes.

Working space

6Problem 6 of 12
Square's diagonals. ABCDABCD is a square with A(0,0)A(0, 0) and C(6,8)C(6, 8).

(a) Find the midpoint of ACAC (the centre of the square).
(b) The diagonals of a square are perpendicular and equal. Find the equation of the other diagonal BDBD.
(c) Hence find BB and DD (given they are symmetric about the centre, each at half the diagonal length from the centre).

Working space

7Problem 7 of 12
Two lines and angle. Line ℓ1\ell_1 has equation y=2x+1y = 2x + 1. Line ℓ2\ell_2 passes through (0,5)(0, 5) and is perpendicular to ℓ1\ell_1.

(a) Find the equation of ℓ2\ell_2.
(b) Find the point of intersection of ℓ1\ell_1 and ℓ2\ell_2.
(c) Find the distance from (0,5)(0, 5) to the intersection point.

Working space

8Problem 8 of 12
Reflection. A line ℓ\ell has equation y=xy = x. Find the image of the point P(3,5)P(3, 5) after reflection in ℓ\ell.

Working space

9Problem 9 of 12
Distance & perimeter problem. A rectangular field has corners at A(0,0)A(0, 0), B(60,0)B(60, 0), C(60,40)C(60, 40), D(0,40)D(0, 40) (measurements in metres).

A diagonal path runs from AA to CC, and another from BB to DD.

(a) Find the length of each diagonal.
(b) Find the coordinates where the two diagonals cross.
(c) A jogger runs around the perimeter once. How far does she run?

Working space

10Problem 10 of 12
System of lines. Two lines pass through the point (2,5)(2, 5). One has gradient 33 and the other has gradient −13-\frac{1}{3}.

(a) Find the equation of each line.
(b) Without sketching, state the angle between them and justify.
(c) The first line crosses the xx-axis at PP and the second crosses the xx-axis at QQ. Find ∣PQ∣|PQ|.

Working space

11Problem 11 of 12
Coordinate proof. Show that the quadrilateral with vertices A(0,0)A(0, 0), B(4,0)B(4, 0), C(5,3)C(5, 3), D(1,3)D(1, 3) is a parallelogram. Is it a rhombus?

Working space

12Problem 12 of 12
Circle through three points (informal). Show that the three points A(0,5)A(0, 5), B(3,4)B(3, 4), C(4,−3)C(4, -3) are all equidistant from the point P(0,0)P(0, 0).

What is the significance of this result?

Working space