Problems — Coordinate Geometry | Mathematics Department

1

**Triangle in the plane.** Points $A(1, 2)$, $B(7, 2)$, and $C(4, 6)$ form a triangle. (a) Find the lengths $AB$, $BC$, $AC$. (b) Is the triangle isosceles, equilateral, or scalene? (c) Find the area of the triangle.

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2

**Parallelogram.** Three vertices of a parallelogram are $A(1, 1)$, $B(5, 3)$, $C(7, 7)$. (a) Find the gradient of $AB$ and $BC$. (b) The fourth vertex $D$ is such that $ABCD$ is a parallelogram (in order). Find $D$. (c) Find the perimeter of the parallelogram in exact form.

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3

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

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4

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

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5

**Find missing coordinates.** A line passes through $(2, -1)$ and has gradient 3. (a) Find its equation. (b) Where does this line cross the $x$-axis and $y$-axis? (c) Find the area of the triangle formed by this line and the two axes.

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6

**Square's diagonals.** $ABCD$ is a square with $A(0, 0)$ and $C(6, 8)$. (a) Find the midpoint of $AC$ (the centre of the square). (b) The diagonals of a square are perpendicular and equal. Find the equation of the other diagonal $BD$. (c) Hence find $B$ and $D$ (given they are symmetric about the centre, each at half the diagonal length from the centre).

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7

**Two lines and angle.** Line $\ell_1$ has equation $y = 2x + 1$. Line $\ell_2$ passes through $(0, 5)$ and is perpendicular to $\ell_1$. (a) Find the equation of $\ell_2$. (b) Find the point of intersection of $\ell_1$ and $\ell_2$. (c) Find the distance from $(0, 5)$ to the intersection point.

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8

**Reflection.** A line $\ell$ has equation $y = x$. Find the image of the point $P(3, 5)$ after reflection in $\ell$.

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9

**Distance & perimeter problem.** A rectangular field has corners at $A(0, 0)$, $B(60, 0)$, $C(60, 40)$, $D(0, 40)$ (measurements in metres). A diagonal path runs from $A$ to $C$, and another from $B$ to $D$. (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?

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10

**System of lines.** Two lines pass through the point $(2, 5)$. One has gradient $3$ and the other has gradient $-\frac{1}{3}$. (a) Find the equation of each line. (b) Without sketching, state the angle between them and justify. (c) The first line crosses the $x$-axis at $P$ and the second crosses the $x$-axis at $Q$. Find $|PQ|$.

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11

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

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12

**Circle through three points (informal).** Show that the three points $A(0, 5)$, $B(3, 4)$, $C(4, -3)$ are all equidistant from the point $P(0, 0)$. What is the significance of this result?

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