Geometries

Coordinate geometry provides a bridge between algebraic equations and geometric figures, enabling the use of algebraic methods to solve geometric problems and vice versa.

2D Coordinate Geometry

The Rectangular Coordinate System

In the Cartesian plane, every point $P$ is uniquely identified by an ordered pair $(x, y)$, where $x$ is the horizontal coordinate (abscissa) and $y$ is the vertical coordinate (ordinate). The axes divide the plane into four quadrants, numbered counterclockwise from the positive $x$-axis.

Distance Formula

The distance between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is derived from the Pythagorean identity applied to the right triangle formed by the horizontal and vertical differences:

$$\begin{aligned} d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \end{aligned}$$

Examples

• Distance between $A(3, 4)$ and $B(7, 1)$: $d = \sqrt{(7-3)^2 + (1-4)^2} = \sqrt{16 + 9} = 5$
• Distance between $O(0, 0)$ and $P(5, 12)$: $d = \sqrt{5^2 + 12^2} = 13$
• Verify that the points $A(0, 0)$, $B(3, 4)$, $C(6, 8)$ are collinear: $AB = 5$, $BC = 5$, $AC = 10$. Since $AB + BC = AC$, the points are collinear.

Midpoint Formula

The midpoint $M$ of the line segment joining $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is:

$$\begin{aligned} M = \left( \frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2} \right) \end{aligned}$$

The midpoint is the average of the respective coordinates of the two endpoints.

Examples

• Midpoint of $A(2, 6)$ and $B(8, -4)$: $M = \left( \frac{2+8}{2}, \frac{6+(-4)}{2} \right) = (5, 1)$
• If $M(4, -1)$ is the midpoint of $A(1, 3)$ and $B$, then $B = (2 \times 4 - 1, 2 \times (-1) - 3) = (7, -5)$

Gradient (Slope)

The gradient $m$ of the line passing through $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, where $x_1 \neq x_2$, measures the rate of change of $y$ with respect to $x$:

$$\begin{aligned} m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta \end{aligned}$$

where $\theta$ is the angle the line makes with the positive $x$-axis. A vertical line (where $x_1 = x_2$) has an undefined gradient. Key gradient values include:

Gradient	Interpretation
$m > 0$	Line slopes upward from left to right
$m < 0$	Line slopes downward from left to right
$m = 0$	Horizontal line ($y = c$)
Undefined	Vertical line ($x = c$)

Examples

• Gradient of the line through $A(1, 2)$ and $B(4, 8)$: $m = \frac{8-2}{4-1} = 2$
• A line with gradient $\frac{3}{4}$ passing through $(1, 5)$: using $y - y_1 = m(x - x_1)$, the equation is $y - 5 = \frac{3}{4}(x - 1)$, i.e., $3x - 4y + 17 = 0$

Division of a Line Segment

If point $P(x, y)$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m : n$ (internally), then:

$$\begin{aligned} x = \frac{mx_2 + nx_1}{m + n}, \qquad y = \frac{my_2 + ny_1}{m + n} \end{aligned}$$

The midpoint formula is the special case where $m = n = 1$.

Equations of Straight Lines

A straight line in the Cartesian plane can be expressed in several equivalent forms.

Slope-Intercept Form

$$\begin{aligned} y = mx + c \end{aligned}$$

where $m$ is the gradient and $c$ is the $y$-intercept. This form is most useful when the gradient and $y$-intercept are known or can be easily determined.

Point-Slope Form

Given a point $(x_1, y_1)$ on the line and the gradient $m$:

$$\begin{aligned} y - y_1 = m(x - x_1) \end{aligned}$$

This is the most versatile form for deriving the equation of a line from partial information.

Two-Point Form

Given two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on the line:

$$\begin{aligned} \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \end{aligned}$$

Intercept Form

If the line has $x$-intercept $a$ and $y$-intercept $b$ (both non-zero):

$$\begin{aligned} \frac{x}{a} + \frac{y}{b} = 1 \end{aligned}$$

General Form

Every straight line can be written in the general form:

$$\begin{aligned} Ax + By + C = 0 \end{aligned}$$

where $A$, $B$, $C$ are constants and $A$ and $B$ are not both zero. The gradient is $m = -\frac{A}{B}$ and the $y$-intercept is $c = -\frac{C}{B}$ (when $B \neq 0$).

Examples

• Find the equation of the line through $(2, 3)$ and $(6, -1)$: $m = \frac{-1 - 3}{6 - 2} = -1$. Equation: $y - 3 = -1(x - 2)$, i.e., $x + y - 5 = 0$
• Find the equation of the line with gradient $\frac{1}{2}$ and $y$-intercept $-3$: $y = \frac{1}{2}x - 3$, i.e., $x - 2y - 6 = 0$
• Convert $3x + 4y - 12 = 0$ to intercept form: $\frac{x}{4} + \frac{y}{3} = 1$. The $x$-intercept is $4$ and the $y$-intercept is $3$.

Parallel and Perpendicular Lines

Parallel Lines

Two non-vertical lines with gradients $m_1$ and $m_2$ are parallel if and only if their gradients are equal:

$$\begin{aligned} m_1 = m_2 \end{aligned}$$

In the general form $Ax + By + C = 0$, two lines $A_1x + B_1y + C_1 = 0$ and $A_2x + B_2y + C_2 = 0$ are parallel if and only if:

$$\begin{aligned} \frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2} \end{aligned}$$

If all three ratios are equal, the lines are coincident (the same line).

Perpendicular Lines

Two non-vertical lines with gradients $m_1$ and $m_2$ are perpendicular if and only if the product of their gradients equals $-1$:

$$\begin{aligned} m_1 \cdot m_2 = -1 \end{aligned}$$

In the general form, two lines are perpendicular if and only if:

$$\begin{aligned} A_1 A_2 + B_1 B_2 = 0 \end{aligned}$$

Examples

• Find the equation of the line through $(1, -2)$ parallel to $2x - 3y + 5 = 0$: The gradient of the given line is $m = \frac{2}{3}$. The parallel line is $y + 2 = \frac{2}{3}(x - 1)$, i.e., $2x - 3y - 8 = 0$.
• Find the equation of the line through $(3, 1)$ perpendicular to $4x + y - 7 = 0$: The gradient of the given line is $m_1 = -4$, so $m_2 = \frac{1}{4}$. The perpendicular line is $y - 1 = \frac{1}{4}(x - 3)$, i.e., $x - 4y + 1 = 0$.
• Determine whether $3x + 2y - 1 = 0$ and $6x + 4y + 5 = 0$ are parallel: Since $\frac{3}{6} = \frac{2}{4} = \frac{1}{2} \neq \frac{-1}{5}$, the lines are parallel but not coincident.

Intersection of Two Lines

The point of intersection of two non-parallel lines $A_1x + B_1y + C_1 = 0$ and $A_2x + B_2y + C_2 = 0$ is found by solving the system simultaneously. The coordinates $(x, y)$ of the intersection satisfy both equations.

Examples

• Find the intersection of $x + y = 5$ and $2x - y = 1$: Adding gives $3x = 6$, so $x = 2$, $y = 3$. The intersection is $(2, 3)$.

Circles

Standard Form of the Equation of a Circle

A circle with centre $C(a, b)$ and radius $r$ has equation:

$$\begin{aligned} (x - a)^2 + (y - b)^2 = r^2 \end{aligned}$$

Expanding this yields the general form:

$$\begin{aligned} x^2 + y^2 - 2ax - 2by + (a^2 + b^2 - r^2) = 0 \end{aligned}$$

General Form of the Equation of a Circle

$$\begin{aligned} x^2 + y^2 + Dx + Ey + F = 0 \end{aligned}$$

where the centre is $\left(-\frac{D}{2}, -\frac{E}{2}\right)$ and the radius is:

$$\begin{aligned} r = \sqrt{\left(\frac{D}{2}\right)^2 + \left(\frac{E}{2}\right)^2 - F} \end{aligned}$$

For a real circle to exist, we require $\left(\frac{D}{2}\right)^2 + \left(\frac{E}{2}\right)^2 - F > 0$.

Examples

• Find the centre and radius of $x^2 + y^2 - 6x + 4y - 12 = 0$: Completing squares, $(x-3)^2 + (y+2)^2 = 25$. Centre $(3, -2)$, radius $5$.
• Find the equation of the circle with centre $(-1, 4)$ and radius $3$: $(x+1)^2 + (y-4)^2 = 9$.
• Find the equation of the circle with diameter endpoints $A(2, 3)$ and $B(8, 7)$: Centre is $M(5, 5)$, radius is $\frac{1}{2}\sqrt{(8-2)^2 + (7-3)^2} = \frac{1}{2}\sqrt{52} = \sqrt{13}$. Equation: $(x-5)^2 + (y-5)^2 = 13$.

Finding the Equation of a Circle from Conditions

The equation of a circle can be determined when given:

• The centre and the radius (direct substitution)
• The centre and a point on the circle (use the distance formula to find $r$)
• Three points on the circle (solve the general form system)
• The endpoints of a diameter (centre is the midpoint, radius is half the distance)

Intersection of a Line and a Circle

To find the points of intersection of the line $y = mx + c$ and the circle $(x-a)^2 + (y-b)^2 = r^2$, substitute the line equation into the circle equation to obtain a quadratic in $x$ (or $y$). The discriminant $\Delta = b^2 - 4ac$ determines the nature of intersection:

Discriminant	Intersection
$\Delta > 0$	Two distinct points (secant)
$\Delta = 0$	One point (tangent)
$\Delta < 0$	No real intersection

Tangent to a Circle

A tangent to a circle at a point $P(x_1, y_1)$ on the circle $(x-a)^2 + (y-b)^2 = r^2$ is perpendicular to the radius at $P$. The gradient of the radius $CP$ (where $C$ is the centre) is:

$$\begin{aligned} m_{\mathrm{radius}} = \frac{y_1 - b}{x_1 - a} \end{aligned}$$

Therefore the gradient of the tangent is:

$$\begin{aligned} m_{\mathrm{tangent}} = -\frac{x_1 - a}{y_1 - b} \quad (y_1 \neq b) \end{aligned}$$

The equation of the tangent at $P(x_1, y_1)$ on the circle $x^2 + y^2 + Dx + Ey + F = 0$ can be obtained by replacing:

$$\begin{aligned} x^2 \to xx_1, \quad y^2 \to yy_1, \quad x \to \frac{x + x_1}{2}, \quad y \to \frac{y + y_1}{2} \end{aligned}$$

This gives:

$$\begin{aligned} xx_1 + yy_1 + \frac{D(x + x_1)}{2} + \frac{E(y + y_1)}{2} + F = 0 \end{aligned}$$

Examples

• Find the equation of the tangent to $x^2 + y^2 = 25$ at $P(3, 4)$: The radius gradient is $\frac{4}{3}$, so the tangent gradient is $-\frac{3}{4}$. Equation: $y - 4 = -\frac{3}{4}(x - 3)$, i.e., $3x + 4y - 25 = 0$. Alternatively, using the formula: $3x + 4y = 25$.
• Determine whether the line $3x - 4y + 10 = 0$ is tangent to $(x-1)^2 + (y+2)^2 = 9$: Substitute $y = \frac{3x+10}{4}$ into the circle. The resulting quadratic has discriminant $\Delta = 0$, confirming tangency.

2D Measurements

Triangles

For a triangle with base $b$ and height $h$:

$$\begin{aligned} \mathrm{Area} &= \frac{1}{2}bh \end{aligned}$$

Using the Heron's formula) for a triangle with side lengths $a$, $b$, $c$ and semi-perimeter $s = \frac{a+b+c}{2}$:

$$\begin{aligned} \mathrm{Area} = \sqrt{s(s-a)(s-b)(s-c)} \end{aligned}$$

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ is given by:

$$\begin{aligned} \mathrm{Area} = \frac{1}{2}\left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \end{aligned}$$

Quadrilaterals

Figure	Area	Perimeter
Rectangle ($l \times w$)	$lw$	$2(l+w)$
Parallelogram ($b$, $h$)	$bh$	$2(a+b)$
Rhombus (diagonals $d_1$, $d_2$)	$\frac{1}{2}d_1 d_2$	$4s$
Trapezium (parallel sides $a$, $b$; height $h$)	$\frac{1}{2}(a+b)h$	Sum of all sides
Square (side $s$)	$s^2$	$4s$

Circles

$$\begin{aligned} \mathrm{Circumference} &= 2\pi r = \pi d \\ \mathrm{Area} &= \pi r^2 \end{aligned}$$

where $r$ is the radius and $d = 2r$ is the diameter.

Arcs and Sectors

For an arc subtending an angle $\theta$ (in radians) at the centre of a circle of radius $r$:

$$\begin{aligned} \mathrm{Arc length} &= l = r\theta \\ \mathrm{Area of sector} &= A = \frac{1}{2}r^2\theta \end{aligned}$$

When the angle is given in degrees ($\theta^\circ$), first convert to radians: $\theta = \theta^\circ \times \frac{\pi}{180}$.

The area of a segment (the region between a chord and the corresponding arc) is:

$$\begin{aligned} \mathrm{Area of segment} = \frac{1}{2}r^2(\theta - \sin\theta) \end{aligned}$$

Examples

• Find the area and perimeter of a sector of radius $7\mathrm{ cm}$ with angle $120^\circ$: $\theta = \frac{2\pi}{3}$ rad. Area $= \frac{1}{2}(49)\left(\frac{2\pi}{3}\right) = \frac{49\pi}{3} \approx 51.3\mathrm{ cm}^2$. Arc length $= 7 \times \frac{2\pi}{3} = \frac{14\pi}{3} \approx 14.7\mathrm{ cm}$. Perimeter $= 14 + \frac{14\pi}{3} \approx 28.7\mathrm{ cm}$.
• A chord of length $8\mathrm{ cm}$ subtends an angle of $90^\circ$ at the centre. Find the area of the minor segment: $r = \frac{8}{\sqrt{2}} = 4\sqrt{2}\mathrm{ cm}$, $\theta = \frac{\pi}{2}$. Area $= \frac{1}{2}(32)\left(\frac{\pi}{2} - 1\right) = 16\left(\frac{\pi}{2} - 1\right) = 8\pi - 16 \approx 9.13\mathrm{ cm}^2$.

3D Geometry

The 3D Coordinate System

In three-dimensional space, each point is identified by an ordered triple $(x, y, z)$. The three coordinate axes --- $x$-axis, $y$-axis, and $z$-axis --- are mutually perpendicular and intersect at the origin $O(0, 0, 0)$. The three planes $xy$-plane, $yz$-plane, and $zx$-plane divide the space into eight octants.

Distance Formula in 3D

The distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in 3D space is:

$$\begin{aligned} d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \end{aligned}$$

The midpoint of the segment $P_1P_2$ is:

$$\begin{aligned} M = \left( \frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2},\; \frac{z_1 + z_2}{2} \right) \end{aligned}$$

Examples

• Distance between $A(1, 2, 3)$ and $B(4, 6, -3)$: $d = \sqrt{3^2 + 4^2 + (-6)^2} = \sqrt{9 + 16 + 36} = \sqrt{61}$
• Show that $A(0, 0, 0)$, $B(1, 0, 0)$, $C(0, 1, 0)$, $D(0, 0, 1)$ form a trirectangular tetrahedron: $AB = AC = AD = 1$ (edges from the origin) and $BC = BD = CD = \sqrt{2}$ (edges of the base triangle $BCD$). It is not a regular tetrahedron — a regular tetrahedron would require all six edges to be equal. For a regular tetrahedron, use vertices such as $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$.

Equation of a Plane

The general equation of a plane in 3D is:

$$\begin{aligned} Ax + By + Cz + D = 0 \end{aligned}$$

where the vector $\mathbf{'\{'}n{'\}'} = (A, B, C)$ is a normal vector to the plane. The plane can also be described in several forms:

• Point-normal form: Given a point $P_0(x_0, y_0, z_0)$ on the plane and normal $\mathbf{'\{'}n{'\}'} = (A, B, C)$:

$$\begin{aligned} A(x - x_0) + B(y - y_0) + C(z - z_0) = 0 \end{aligned}$$

• Intercept form: If the plane has $x$-intercept $a$, $y$-intercept $b$, and $z$-intercept $c$ (all non-zero):

$$\begin{aligned} \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \end{aligned}$$

Angles Between Lines and Planes

Angle Between a Line and a Plane

The angle $\phi$ between a line with direction vector $\mathbf{'\{'}d{'\}'} = (l, m, n)$ and a plane with normal $\mathbf{'\{'}n{'\}'} = (A, B, C)$ is defined as the complement of the angle between $\mathbf{'\{'}d{'\}'}$ and $\mathbf{'\{'}n{'\}'}$:

$$\begin{aligned} \sin\phi = \frac{|Al + Bm + Cn|}{\sqrt{A^2 + B^2 + C^2}\sqrt{l^2 + m^2 + n^2}} \end{aligned}$$

Angle Between Two Planes

The angle $\theta$ between two planes with normals $\mathbf{'\{'}n_1{'\}'} = (A_1, B_1, C_1)$ and $\mathbf{'\{'}n_2{'\}'} = (A_2, B_2, C_2)$ is:

$$\begin{aligned} \cos\theta = \frac{|A_1 A_2 + B_1 B_2 + C_1 C_2|}{\sqrt{A_1^2 + B_1^2 + C_1^2}\sqrt{A_2^2 + B_2^2 + C_2^2}} \end{aligned}$$

Angle Between Two Lines in 3D

For two lines with direction vectors $\mathbf{'\{'}d_1{'\}'} = (l_1, m_1, n_1)$ and $\mathbf{'\{'}d_2{'\}'} = (l_2, m_2, n_2)$:

$$\begin{aligned} \cos\theta = \frac{|l_1 l_2 + m_1 m_2 + n_1 n_2|}{\sqrt{l_1^2 + m_1^2 + n_1^2}\sqrt{l_2^2 + m_2^2 + n_2^2}} \end{aligned}$$

Distance from a Point to a Plane

The perpendicular distance from point $P(x_0, y_0, z_0)$ to the plane $Ax + By + Cz + D = 0$ is:

$$\begin{aligned} d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \end{aligned}$$

Examples

• Find the distance from $P(1, 2, -1)$ to the plane $2x - y + 2z + 3 = 0$: $d = \frac{|2(1) - 1(2) + 2(-1) + 3|}{\sqrt{4 + 1 + 4}} = \frac{|2 - 2 - 2 + 3|}{3} = \frac{1}{3}$.
• Find the angle between the planes $x + y + z = 1$ and $2x - y + z = 0$: $\cos\theta = \frac{|1\cdot 2 + 1\cdot(-1) + 1\cdot 1|}{\sqrt{3}\sqrt{6}} = \frac{2}{3\sqrt{2}} = \frac{\sqrt{2}}{3}$. So $\theta = \arccos\left(\frac{\sqrt{2}}{3}\right) \approx 61.9^\circ$.
• Find the equation of the plane through $(1, 0, 2)$, $(0, 1, -1)$, and $(2, 1, 1)$: direction vectors $\overrightarrow{AB} = (-1, 1, -3)$ and $\overrightarrow{AC} = (1, 1, -1)$. Cross product: $\overrightarrow{AB} \times \overrightarrow{AC} = (2, -4, -2)$, so normal $\mathbf{'\{'}n{'\}'} = (1, -2, -1)$. The plane is $x - 2y - z = -1$.

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Wrap-up Questions

1. Question: Find the equation of the perpendicular bisector of the line segment joining $A(3, -1)$ and $B(7, 5)$.

Answer

• Midpoint: $M = \left(\frac{3+7}{2}, \frac{-1+5}{2}\right) = (5, 2)$.
• Gradient of $AB$: $m_{AB} = \frac{5 - (-1)}{7 - 3} = \frac{3}{2}$.
• Gradient of perpendicular bisector: $m = -\frac{2}{3}$.
• Equation: $y - 2 = -\frac{2}{3}(x - 5)$, i.e., $2x + 3y - 16 = 0$.

2. Question: Find the equation of the circle passing through the three points $A(0, 0)$, $B(4, 0)$, and $C(0, 3)$.

Answer

• Let the circle be $x^2 + y^2 + Dx + Ey + F = 0$.
• Substituting $A(0,0)$: $F = 0$.
• Substituting $B(4,0)$: $16 + 4D = 0 \implies D = -4$.
• Substituting $C(0,3)$: $9 + 3E = 0 \implies E = -3$.
• Equation: $x^2 + y^2 - 4x - 3y = 0$, i.e., $(x-2)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{25}{4}$.
• Centre $\left(2, \frac{3}{2}\right)$, radius $\frac{5}{2}$.

3. Question: The line $y = 2x + k$ is tangent to the circle $x^2 + y^2 - 4x - 2y + 1 = 0$. Find the value(s) of $k$.

Answer

• Substitute $y = 2x + k$ into $(x-2)^2 + (y-1)^2 = 4$:
• $(x-2)^2 + (2x + k - 1)^2 = 4$.
• Expanding: $x^2 - 4x + 4 + 4x^2 + 4(k-1)x + (k-1)^2 = 4$.
• $5x^2 + (4k - 8)x + (k^2 - 2k + 1) = 0$.
• For tangency, $\Delta = 0$: $(4k-8)^2 - 4(5)(k^2 - 2k + 1) = 0$.
• $16k^2 - 64k + 64 - 20k^2 + 40k - 20 = 0$.
• $-4k^2 - 24k + 44 = 0 \implies k^2 + 6k - 11 = 0$.
• $k = \frac{-6 \pm \sqrt{36 + 44}}{2} = -3 \pm \sqrt{14}$.

4. Question: A sector of a circle of radius $12\mathrm{ cm}$ has an arc length of $16\pi\mathrm{ cm}$. Find the area of the sector and the area of the corresponding segment if the chord length is $20\mathrm{ cm}$.

Answer

• Arc length $l = r\theta$: $16\pi = 12\theta \implies \theta = \frac{4\pi}{3}$ rad.
• Area of sector: $A = \frac{1}{2}r^2\theta = \frac{1}{2}(144)\left(\frac{4\pi}{3}\right) = 96\pi \mathrm{ cm}^2$.
• Area of triangle formed by the radii and chord: Using the chord length $c = 20$, the triangle has sides $12$, $12$, $20$. Semi-perimeter $s = 22$. By Heron's formula: Area $= \sqrt{22 \times 10 \times 10 \times 2} = \sqrt{4400} = 20\sqrt{11} \mathrm{ cm}^2$.
• Alternatively, the perpendicular from centre to chord: $h = \sqrt{12^2 - 10^2} = 2\sqrt{11}$. Triangle area $= \frac{1}{2} \times 20 \times 2\sqrt{11} = 20\sqrt{11}$.
• Area of segment $= 96\pi - 20\sqrt{11} \approx 247.3 \mathrm{ cm}^2$.

5. Question: Find the angle between the line joining $A(1, 2, 3)$ and $B(4, 5, 6)$ and the plane $x + y + z = 1$.

Answer

• Direction vector of the line: $\mathbf{'\{'}d{'\}'} = (4-1, 5-2, 6-3) = (3, 3, 3)$, simplified to $(1, 1, 1)$.
• Normal of the plane: $\mathbf{'\{'}n{'\}'} = (1, 1, 1)$.
• The angle $\alpha$ between $\mathbf{'\{'}d{'\}'}$ and $\mathbf{'\{'}n{'\}'}$: $\cos\alpha = \frac{|1+1+1|}{\sqrt{3}\sqrt{3}} = \frac{3}{3} = 1$, so $\alpha = 0^\circ$.
• The angle $\phi$ between the line and the plane is the complement: $\phi = 90^\circ - 0^\circ = 90^\circ$.
• The line is parallel to the plane (perpendicular to the normal).

6. Question: Points $A(1, 2)$, $B(4, 6)$, and $C(7, 4)$ are vertices of a triangle. Find the equation of the altitude from $A$ to $BC$, the area of triangle $ABC$, and the length of the altitude from $A$.

Answer

• Gradient of $BC$: $m_{BC} = \frac{4-6}{7-4} = -\frac{2}{3}$.
• Gradient of altitude from $A$: $m = \frac{3}{2}$ (negative reciprocal).
• Equation of altitude: $y - 2 = \frac{3}{2}(x - 1)$, i.e., $3x - 2y + 1 = 0$.
• Area of $\triangle ABC$: $\frac{1}{2}|1(6-4) + 4(4-2) + 7(2-6)| = \frac{1}{2}|2 + 8 - 28| = \frac{1}{2}(18) = 9$ square units.
• Length of $BC$: $\sqrt{(7-4)^2 + (4-6)^2} = \sqrt{9 + 4} = \sqrt{13}$.
• Altitude from $A$: $\frac{2 \times \mathrm{Area}}{BC} = \frac{18}{\sqrt{13}} = \frac{18\sqrt{13}}{13}$.

7. Question: Find the equation of the circle which touches the $x$-axis at $(3, 0)$ and passes through $(1, 4)$.

Answer

• Since the circle touches the $x$-axis at $(3, 0)$, the centre lies on the vertical line $x = 3$, so the centre is $C(3, r)$ where $r$ is the radius.
• The circle equation is $(x-3)^2 + (y-r)^2 = r^2$.
• Substituting $(1, 4)$: $(1-3)^2 + (4-r)^2 = r^2$.
• $4 + 16 - 8r + r^2 = r^2 \implies 20 - 8r = 0 \implies r = \frac{5}{2}$.
• Centre: $(3, \frac{5}{2})$, radius: $\frac{5}{2}$.
• Equation: $(x-3)^2 + \left(y - \frac{5}{2}\right)^2 = \frac{25}{4}$, i.e., $x^2 + y^2 - 6x - 5y + 9 = 0$.

8. Question: The vertices of a triangle are $A(2, 1, -1)$, $B(0, 3, -4)$, and $C(5, 0, 2)$. Find (a) the length of each side, (b) the area of the triangle, and (c) the equation of the plane containing the triangle.

Answer

• (a) Side lengths:
  • $AB = \sqrt{(0-2)^2 + (3-1)^2 + (-4+1)^2} = \sqrt{4 + 4 + 9} = \sqrt{17}$
  • $BC = \sqrt{(5-0)^2 + (0-3)^2 + (2+4)^2} = \sqrt{25 + 9 + 36} = \sqrt{70}$
  • $CA = \sqrt{(2-5)^2 + (1-0)^2 + (-1-2)^2} = \sqrt{9 + 1 + 9} = \sqrt{19}$
• (b) Using the cross product method: $\overrightarrow{AB} = (-2, 2, -3)$, $\overrightarrow{AC} = (3, -1, 3)$.
  • $\overrightarrow{AB} \times \overrightarrow{AC} = (2 \cdot 3 - (-3)(-1),\; (-3) \cdot 3 - (-2) \cdot 3,\; (-2)(-1) - 2 \cdot 3) = (3, -3, -4)$.
  • Area $= \frac{1}{2}|\overrightarrow{AB} \times \overrightarrow{AC}| = \frac{1}{2}\sqrt{9 + 9 + 16} = \frac{\sqrt{34}}{2}$.
• (c) The normal vector is $(3, -3, -4)$. Using point $A(2, 1, -1)$: $3(x-2) - 3(y-1) - 4(z+1) = 0$, i.e., $3x - 3y - 4z - 8 = 0$.

9. Question: Two circles $C_1: x^2 + y^2 - 4x + 2y - 4 = 0$ and $C_2: x^2 + y^2 + 2x - 6y + 6 = 0$ intersect at points $A$ and $B$. Find the equation of the common chord $AB$ and the coordinates of $A$ and $B$.

Answer

• Common chord: Subtract the two equations: $(-4x + 2y - 4) - (2x - 6y + 6) = 0$.
• $-6x + 8y - 10 = 0$, i.e., $3x - 4y + 5 = 0$.
• From $C_1$: $(x-2)^2 + (y+1)^2 = 9$. Substitute $y = \frac{3x+5}{4}$:
• $(x-2)^2 + \left(\frac{3x+5}{4} + 1\right)^2 = 9$.
• $(x-2)^2 + \left(\frac{3x+9}{4}\right)^2 = 9$.
• Multiply by $16$: $16(x-2)^2 + (3x+9)^2 = 144$.
• $16(x^2 - 4x + 4) + 9x^2 + 54x + 81 = 144$.
• $25x^2 - 10x + 145 = 144$.
• $25x^2 - 10x + 1 = 0$.
• Discriminant $= 100 - 100 = 0$, so the circles are tangent (touch at exactly one point).
• $x = \frac{10}{50} = \frac{1}{5}$, $y = \frac{3(1/5) + 5}{4} = \frac{28}{20} = \frac{7}{5}$.
• The circles touch at $\left(\frac{1}{5}, \frac{7}{5}\right)$.

10. Question: A rectangle $ABCD$ has vertices $A(1, 1)$, $B(5, 1)$, and $C(5, 4)$. Find the coordinates of $D$ and the area of the rectangle.

Answer

• $AB$ is horizontal: $AB = 5 - 1 = 4$. $BC$ is vertical: $BC = 4 - 1 = 3$.
• Verify: $AB^2 + BC^2 = 16 + 9 = 25 = AC^2$, so $\angle ABC = 90^\circ$. ✓
• The diagonal is $AC$ with midpoint $M = \left(\frac{1+5}{2}, \frac{1+4}{2}\right) = (3, 2.5)$.
• $D$ is such that $M$ is also the midpoint of $CD$: $D = (2M_x - C_x, 2M_y - C_y) = (6 - 5, 5 - 4) = (1, 4)$.
• Area $= AB \times BC = 4 \times 3 = 12$ square units.

11. Question: A solid metal cone has base radius $6\mathrm{ cm}$ and slant height $10\mathrm{ cm}$. A sector is cut from a circular sheet of metal to form the curved surface. Find (a) the angle of the sector, (b) the radius of the circular sheet, and (c) the area of the sector.

Answer

• (a) The arc length of the sector equals the circumference of the base: $l = 2\pi(6) = 12\pi$.
• The sector radius equals the slant height: $r = 10\mathrm{ cm}$.
• $l = r\theta \implies 12\pi = 10\theta \implies \theta = \frac{6\pi}{5}$ rad $= 216^\circ$.
• (b) The radius of the circular sheet is the slant height: $10\mathrm{ cm}$.
• (c) Area of sector: $A = \frac{1}{2}r^2\theta = \frac{1}{2}(100)\left(\frac{6\pi}{5}\right) = 60\pi \mathrm{ cm}^2$.

12. Question: Prove that the points $A(-2, 0)$, $B(0, 4)$, $C(6, 2)$, and $D(4, -2)$ are the vertices of a rectangle. Find the equation of the circle passing through all four vertices.

Answer

• Midpoint of $AC = \left(\frac{-2+6}{2}, \frac{0+2}{2}\right) = (2, 1)$.
• Midpoint of $BD = \left(\frac{0+4}{2}, \frac{4+(-2)}{2}\right) = (2, 1)$.
• Since the diagonals bisect each other, $ABCD$ is a parallelogram.
• $AB = \sqrt{4 + 16} = 2\sqrt{5}$, $BC = \sqrt{36 + 4} = 2\sqrt{10}$, $AC = \sqrt{64 + 4} = 2\sqrt{17}$.
• $AB^2 + BC^2 = 20 + 40 = 60 = AC^2$. The parallelogram has a right angle, so $ABCD$ is a rectangle.
• The circle through all four vertices (circumcircle) has centre at the intersection of the diagonals $(2, 1)$ and radius $= \frac{1}{2}AC = \sqrt{17}$.
• Equation: $(x-2)^2 + (y-1)^2 = 17$, i.e., $x^2 + y^2 - 4x - 2y - 12 = 0$.

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tip

Diagnostic Test Ready to test your understanding of Geometries? The diagnostic test contains the hardest questions within the DSE specification for this topic, each with a full worked solution.

Unit tests probe edge cases and common misconceptions. Integration tests combine Geometries with other DSE mathematics topics to test synthesis under exam conditions.

See Diagnostic Guide for instructions on self-marking and building a personal test matrix.

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DSE Exam Technique

Showing Working

For geometry problems in DSE Paper 1:

1. When using the distance or midpoint formula, write the formula before substituting.
2. When finding the equation of a line, state the gradient and a point, then apply the point-slope form.
3. For circle problems, show the completing-the-square steps to find centre and radius.
4. For 3D problems, clearly identify the right-angled triangles used and label the diagram.
5. For area problems, show which formula is being used and justify the values substituted.

Significant Figures

Length answers to 3 significant figures unless exact. Angle answers to nearest degree or 3 s.f. as appropriate.

Common DSE Question Types

1. Circle equations (finding centre, radius, tangent equations).
2. 2D area problems (shoelace formula, Heron's formula).
3. 3D distance and angle problems.
4. Arc length and sector area calculations.
5. Parallel/perpendicular line problems.

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Additional Worked Examples

Worked Example 13: Two circles touching externally

Circle $C_1$ has centre $(1, 2)$ and radius $3$. Circle $C_2$ has centre $(7, 2)$ and radius $r$. If $C_1$ and $C_2$ touch externally, find $r$.

Solution

For external tangency, the distance between centres equals the sum of the radii:

$d = \sqrt{(7 - 1)^2 + (2 - 2)^2} = 6$

$3 + r = 6 \implies r = 3$

Worked Example 14: Area of a regular hexagon

A regular hexagon has side length $a$. Find its area in terms of $a$.

Solution

A regular hexagon can be divided into 6 equilateral triangles, each with side $a$.

Area of one equilateral triangle: $\dfrac{\sqrt{3}}{4}a^2$.

Total area: $6 \times \dfrac{\sqrt{3}}{4}a^2 = \dfrac{3\sqrt{3}}{2}a^2$.

Worked Example 15: Shortest distance between skew lines (3D)

In 3D, find the distance between the point $P(1, 2, 3)$ and the line through $A(4, 5, 6)$ and $B(7, 8, 9)$.

Solution

Direction vector of the line: $\mathbf{'\{'}d{'\}'} = (3, 3, 3)$, simplified to $(1, 1, 1)$.

Vector $\overrightarrow{AP} = (1 - 4, 2 - 5, 3 - 6) = (-3, -3, -3)$.

The cross product $\overrightarrow{AP} \times \mathbf{'\{'}d{'\}'} = (-3, -3, -3) \times (1, 1, 1)$.

Since $\overrightarrow{AP}$ is parallel to $\mathbf{'\{'}d{'\}'}$ (one is $-3$ times the other), the point $P$ lies on the line.

The distance is $0$.

Verification: $P$ lies on the line through $A$ in direction $(1,1,1)$ since $(1, 2, 3) = (4, 5, 6) + (-1)(1, 1, 1)$. So $P$ is on the line and the distance is $0$.

Worked Example 16: 3D tetrahedron volume

Find the volume of the tetrahedron with vertices $A(0, 0, 0)$, $B(2, 0, 0)$, $C(0, 3, 0)$, $D(0, 0, 4)$.

Solution

The volume of a tetrahedron with vertices at the origin and on the coordinate axes is:

$V = \frac{1}{6}|x_B \cdot y_C \cdot z_D| = \frac{1}{6}(2)(3)(4) = 4$

Alternatively, using the scalar triple product:

$V = \frac{1}{6}|\overrightarrow{AB} \cdot (\overrightarrow{AC} \times \overrightarrow{AD})| = \frac{1}{6}|(2, 0, 0) \cdot ((0, 3, 0) \times (0, 0, 4))|$

$(0, 3, 0) \times (0, 0, 4) = (12, 0, 0)$

$V = \frac{1}{6}|(2)(12)| = \frac{1}{6}(24) = 4$

Worked Example 17: Angle between two lines in 3D

Find the angle between the lines joining $A(1, 0, 0)$ to $B(0, 1, 0)$ and $A$ to $C(0, 0, 1)$.

Solution

$\overrightarrow{AB} = (-1, 1, 0)$, $\overrightarrow{AC} = (-1, 0, 1)$.

$\cos\theta = \frac{|(-1)(-1) + (1)(0) + (0)(1)|}{\sqrt{1 + 1 + 0} \cdot \sqrt{1 + 0 + 1}} = \frac{1}{\sqrt{2}\sqrt{2}} = \frac{1}{2}$

$\theta = 60^\circ$

Worked Example 18: Equation of tangent to a circle with general form

Find the equation of the tangent to $x^2 + y^2 - 4x + 6y - 12 = 0$ at the point $(5, 1)$.

Solution

First verify $(5, 1)$ lies on the circle: $25 + 1 - 20 + 6 - 12 = 0$. Confirmed.

Using the tangent formula for the general circle equation $x^2 + y^2 + Dx + Ey + F = 0$:

Replace $x^2 \to x \cdot 5$, $y^2 \to y \cdot 1$, $x \to \dfrac{x + 5}{2}$, $y \to \dfrac{y + 1}{2}$:

$5x + y - 4\cdot\frac{x + 5}{2} + 6\cdot\frac{y + 1}{2} - 12 = 0$

$5x + y - 2x - 10 + 3y + 3 - 12 = 0$

$3x + 4y - 19 = 0$

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DSE Exam-Style Questions

DSE Practice 1. Find the equation of the circle which passes through $(0, 0)$ and $(0, 4)$ and has its centre on the line $x - y + 2 = 0$.

Solution

Let the centre be $C(h, k)$. Since $C$ lies on $x - y + 2 = 0$: $h - k + 2 = 0 \implies k = h + 2$.

$CA^2 = CB^2$ where $A = (0, 0)$ and $B = (0, 4)$:

$h^2 + k^2 = h^2 + (k - 4)^2 \implies k^2 = k^2 - 8k + 16 \implies 8k = 16 \implies k = 2$

$h = k - 2 = 0$.

Centre: $(0, 2)$. $r^2 = 0^2 + 2^2 = 4$.

Equation: $x^2 + (y - 2)^2 = 4$, or $x^2 + y^2 - 4y = 0$.

DSE Practice 2. In $\triangle ABC$, $A = (1, -2)$, $B = (5, 4)$, and $C = (8, 1)$. Find the equation of the median from $A$ and the equation of the altitude from $B$.

Solution

Median from $A$: Midpoint of $BC$: $M = \left(\dfrac{5+8}{2}, \dfrac{4+1}{2}\right) = (6.5, 2.5)$.

Gradient of $AM$: $m = \dfrac{2.5 - (-2)}{6.5 - 1} = \dfrac{4.5}{5.5} = \dfrac{9}{11}$.

Equation: $y + 2 = \dfrac{9}{11}(x - 1) \implies 11y + 22 = 9x - 9 \implies 9x - 11y - 31 = 0$.

Altitude from $B$: Gradient of $AC$: $m_{AC} = \dfrac{1 - (-2)}{8 - 1} = \dfrac{3}{7}$.

Gradient of altitude: $m_\perp = -\dfrac{7}{3}$.

Equation through $B(5, 4)$: $y - 4 = -\dfrac{7}{3}(x - 5) \implies 3y - 12 = -7x + 35 \implies 7x + 3y - 47 = 0$.

DSE Practice 3. A sector of a circle of radius $10\mathrm{ cm}$ has perimeter $30\mathrm{ cm}$. Find the area of the sector.

Solution

Perimeter of sector $= 2r + r\theta = 20 + 10\theta = 30$.

$10\theta = 10 \implies \theta = 1$ rad.

$\mathrm{Area} = \frac{1}{2}r^2\theta = \frac{1}{2}(100)(1) = 50\mathrm{ cm}^2$

DSE Practice 4. Two circles $C_1: (x - 1)^2 + y^2 = 9$ and $C_2: (x - 5)^2 + y^2 = 25$ intersect at $A$ and $B$. Find the length of the common chord $AB$.

Solution

Centres: $O_1 = (1, 0)$, $O_2 = (5, 0)$. Radii: $r_1 = 3$, $r_2 = 5$.

Distance between centres: $d = 4$.

The common chord is perpendicular to $O_1O_2$ (the $x$-axis) and passes through the midpoint.

By symmetry, the common chord is the line $x = c$ for some $c$.

Substituting $x = c$ into $C_1$: $(c - 1)^2 + y^2 = 9 \implies y^2 = 9 - (c - 1)^2$.

Substituting $x = c$ into $C_2$: $(c - 5)^2 + y^2 = 25 \implies y^2 = 25 - (c - 5)^2$.

Setting equal: $9 - (c - 1)^2 = 25 - (c - 5)^2$.

$9 - c^2 + 2c - 1 = 25 - c^2 + 10c - 25$

$8 + 2c = 10c \implies 8c = 8 \implies c = 1$.

$y^2 = 9 - 0 = 9 \implies y = \pm 3$. Points: $(1, 3)$ and $(1, -3)$.

$AB = 6$.

DSE Practice 5. Find the angle between the planes $2x - y + 2z = 3$ and $x + 2y - 2z = 1$.

Solution

Normal to first plane: $\mathbf{'\{'}n_1{'\}'} = (2, -1, 2)$. Normal to second plane: $\mathbf{'\{'}n_2{'\}'} = (1, 2, -2)$.

$\cos\theta = \frac{|2(1) + (-1)(2) + 2(-2)|}{\sqrt{4 + 1 + 4} \cdot \sqrt{1 + 4 + 4}} = \frac{|2 - 2 - 4|}{3 \cdot 3} = \frac{4}{9}$

$\theta = \arccos\!\left(\frac{4}{9}\right) \approx 63.6^\circ$

DSE Practice 6. A pyramid has a rectangular base $ABCD$ with $AB = 8$, $BC = 6$. The vertex $V$ is directly above the centre of the base, and $VA = 10$. Find the angle between the face $VAB$ and the base $ABCD$.

Solution

The centre of the rectangle is $O = (4, 3, 0)$ (taking $A$ at the origin).

$VA = 10$, $OA = \sqrt{4^2 + 3^2} = 5$.

$VO = \sqrt{VA^2 - OA^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3}$.

Height of the pyramid: $h = 5\sqrt{3}$.

The midpoint of $AB$ is $M = (4, 0, 0)$.

$VM = \sqrt{VO^2 + OM^2} = \sqrt{75 + 9} = \sqrt{84} = 2\sqrt{21}$.

The angle between face $VAB$ and the base is the angle between $VM$ and $OM$:

$\cos\phi = \frac{OM}{VM} = \frac{3}{2\sqrt{21}} = \frac{\sqrt{21}}{14}$

$\phi = \arccos\!\left(\frac{\sqrt{21}}{14}\right) \approx 69.2^\circ$

DSE Practice 7. The line $3x + 4y - 25 = 0$ intersects the circle $x^2 + y^2 = 25$ at $A$ and $B$. Find the area of the minor segment cut off by the chord $AB$.

Solution

Distance from the centre $(0, 0)$ to the line:

$d = \frac{|25|}{5} = 5$

Since $d = r = 5$, the line is tangent to the circle (not a chord). There is no minor segment -- the line touches the circle at exactly one point.

The point of contact: the foot of the perpendicular from $(0,0)$ to $3x + 4y = 25$.

$3x + 4y = 25$ and $y = \dfrac{3}{4}x$ (the perpendicular from the origin).

$3x + 4 \cdot \dfrac{3}{4}x = 25 \implies 6x = 25 \implies x = \dfrac{25}{6}$, $y = \dfrac{25}{8}$.

Point of tangency: $\left(\dfrac{25}{6}, \dfrac{25}{8}\right)$.

Since the line is tangent (not a secant), the area of the minor segment is $0$.