Space

The One Thing

Space mathematics combines geometric reasoning with coordinate methods, Pythagoras’ theorem and trigonometry to describe, justify and solve problems involving shapes, positions, lengths and angles. Successful solutions depend on selecting a theorem or formula whose conditions are satisfied, retaining exact values where practical, and checking the result against the diagram and context.

Definitions and Results

• Point: An exact location on a plane, represented by an ordered pair of coordinates such as $(x,y)$.
• Cartesian plane: A coordinate system formed by a horizontal $x$-axis and vertical $y$-axis, divided into four quadrants.
• Gradient: The steepness and direction of a line:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$ This formula requires $x_2\ne x_1$.

• Distance between two points: The straight-line distance between $(x_1,y_1)$ and $(x_2,y_2)$:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$ This formula follows from Pythagoras’ theorem.

• Midpoint: The point halfway between two coordinates:

$$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

• Equation of a straight line: A rule describing all points on a line, commonly written:

$$y=mx+b,$$ where $m$ is the gradient and $b$ is the $y$-intercept. It may also be written in point-gradient form: $$y-y_1=m(x-x_1).$$

• Geometric proof: A logical sequence of statements establishing why a geometric result must be true.
• Congruent shapes: Shapes with the same size and shape, even if one has been rotated, reflected or translated.
• Similar shapes: Shapes with equal corresponding angles and proportional corresponding side lengths.
• Parallel lines: Lines with equal gradients.
• Perpendicular lines: For non-vertical perpendicular lines, their gradients satisfy:

$$m_1{m}_2=-1.$$

• Horizontal line: A line with gradient $0$.
• Vertical line: A line with undefined gradient.
• Pythagoras’ theorem: For a right-angled triangle:

$$c^2=a^2+b^2,$$ where $c$ is the hypotenuse.

• Hypotenuse: The side opposite the right angle in a right-angled triangle; it is always the longest side.
• Shorter side using Pythagoras: If $c$ is the hypotenuse:

$$a^2=c^2-b^2,$$ followed by taking the positive square root.

• Converse of Pythagoras’ theorem: If $c^2=a^2+b^2$, then the angle opposite $c$ is $90^{\circ }$.
• Sine: For a chosen angle $\theta$:

$$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}.$$

• Cosine: For a chosen angle $\theta$:

$$\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}.$$

• Tangent: For a chosen angle $\theta$:

$$\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}.$$

• Inverse trigonometric function: A calculator function such as ${\sin }^{-1}$, ${\cos }^{-1}$ or ${\tan }^{-1}$ used to find an unknown angle:

$$\theta ={\sin }^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right),$$ $$\theta ={\cos }^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right),$$ or $$\theta ={\tan }^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right).$$ The calculator should be set to degree mode when angles are measured in degrees.

• Angle of elevation: The angle measured upwards from a horizontal line to an object above the observer.
• Angle of depression: The angle measured downwards from a horizontal line to an object below the observer.
• Bearing: A direction measured clockwise from north, usually written as a three-digit angle, such as $135^{\circ }$.
• Exact value: A value expressed without rounding, often using fractions or surds such as $\sqrt{2}$.
• Surd: An expression involving an irrational root that cannot be simplified to a rational number, such as $\sqrt{3}$.
• Scale diagram: A drawing in which lengths are proportional to actual lengths according to a stated scale.
• Three-dimensional geometry: The study of objects with length, width and height, including distances and angles in solids.
• Angle facts: Angles in a triangle add to $180^{\circ }$; angles around a point add to $360^{\circ }$; and vertically opposite angles are equal.
• Parallel-line angle facts: For parallel lines cut by a transversal, corresponding and alternate interior angles are equal, while co-interior angles add to $180^{\circ }$.
• Interior angles of a polygon: The sum of the interior angles of an $n$-sided polygon is:

$$(n-2)\times 180^{\circ }.$$

• Exterior angles of a convex polygon: The exterior angles add to:

$$360^{\circ }.$$

• Area of a triangle:

$$A=\frac{1}{2}\times \text{base}\times \text{perpendicular height}.$$

• Circumference of a circle:

$$C=2\pi r.$$

• Area of a circle:

$$A=\pi r^2.$$

Worked Methods

1. Using coordinate geometry

• Identify the two points or the relevant line information on the Cartesian plane.
• To calculate the gradient between $(x_1,y_1)$ and $(x_2,y_2)$, use:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Check that $x_2\ne x_1$. If the line is vertical, its gradient is undefined.
• To calculate the distance between the points, use:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$ This is obtained by applying Pythagoras’ theorem to the horizontal and vertical changes.

• To find the midpoint, use:

$$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

• To determine whether lines are parallel, compare their gradients. Equal gradients indicate parallel lines.
• To determine whether non-vertical lines are perpendicular, verify:

$$m_1{m}_2=-1.$$

• To construct the equation of a line, substitute the gradient and $y$-intercept into:

$$y=mx+b,$$ or use: $$y-y_1=m(x-x_1)$$ when a point and gradient are known.

2. Applying geometric reasoning and angle properties

• Identify the definitions, known angle facts, congruence, similarity or parallel-line relationships relevant to the diagram.
• State the known information precisely.
• Use the appropriate result:

- angles in a triangle add to $180^{\circ }$; - angles around a point add to $360^{\circ }$; - vertically opposite angles are equal; - corresponding and alternate interior angles are equal for parallel lines; - co-interior angles add to $180^{\circ }$.

• For polygons, use:

$$(n-2)\times 180^{\circ }$$ for the sum of interior angles, or $360^{\circ }$ for the exterior angles of any convex polygon.

• Present the conclusion as a logical sequence rather than relying on the apparent appearance of the diagram.
• Where appropriate, establish that shapes are congruent or similar using their defining properties.

3. Applying Pythagoras’ theorem

• Identify a $90^{\circ }$ angle. Pythagoras’ theorem applies only to right-angled triangles.
• Identify the hypotenuse, which is opposite the right angle and is the longest side.
• Label the other two sides $a$ and $b$, and the hypotenuse $c$.
• To find the hypotenuse, use:

$$c^2=a^2+b^2.$$

• To find a shorter side, rearrange:

$$a^2=c^2-b^2$$ and take the positive square root.

• Retain exact values during working where practical and round only at the final step.
• Check that the answer is reasonable and that the hypotenuse is the longest side.
• To test whether a triangle is right-angled, identify the longest side $c$ and check whether:

$$c^2=a^2+b^2.$$ If this equality holds, the angle opposite $c$ is $90^{\circ }$.

• In three-dimensional geometry, a diagonal across a rectangular prism can often be found by applying Pythagoras twice: first on one face, then using the resulting diagonal with the remaining dimension.

4. Applying trigonometric ratios

• Identify the chosen angle in the right-angled triangle.
• Identify the three sides relative to that angle:

- opposite; - adjacent; - hypotenuse.

• Select the ratio containing the known and unknown sides:

$$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}},$$ $$\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}},$$ $$\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}.$$

• Substitute the known values and solve for the unknown side.
• If the unknown is an angle, use the appropriate inverse function:

$$\theta ={\sin }^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right),$$ $$\theta ={\cos }^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right),$$ or $$\theta ={\tan }^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right).$$

• Use degree mode when the problem measures angles in degrees.
• Round suitably at the final step and check the result against the diagram.

5. Solving practical and three-dimensional problems

• Interpret the diagram as a model, identifying its scale, units, known lengths and relevant angles.
• Identify the right angle or reference angle.
• Label the known sides and determine whether the problem requires Pythagoras, trigonometry, coordinate geometry, an area formula or a circumference formula.
• Calculate using the selected theorem or formula.
• For a rectangular prism diagonal, apply Pythagoras first to a face and then apply it again using the resulting face diagonal and the remaining dimension.
• Retain exact answers where practical, including fractions or surds such as $\sqrt{2}$ and $\sqrt{3}$.
• Round usually only at the final step.
• Check the units, scale and whether the result is reasonable in relation to the diagram and real situation.

Where It Goes Wrong

• Applying Pythagoras without first identifying a $90^{\circ }$ angle; the theorem applies only to right-angled triangles.
• Misidentifying the hypotenuse; it is opposite the right angle and is always the longest side.
• Using the gradient formula when $x_2=x_1$; a vertical line has undefined gradient, not gradient $0$.
• Treating a line as parallel or perpendicular from its appearance instead of checking gradients: parallel lines have equal gradients, while non-vertical perpendicular lines satisfy $m_1{m}_2=-1$.
• Choosing sine, cosine or tangent without identifying the selected angle and the opposite, adjacent and hypotenuse sides.
• Using an inverse trigonometric function with the calculator in radian mode when the angles are given in degrees, or rounding too early instead of retaining exact values until the final step.

What Gets Asked

This material supports questions requiring students to:

• calculate the gradient, distance and midpoint between two points on the Cartesian plane;
• find the equation of a straight line in the form $y=mx+b$ or point-gradient form;
• test whether lines are parallel or perpendicular;
• use coordinate relationships to verify lengths, intersections and symmetry;
• justify geometric conclusions using definitions, angle facts, congruence or similarity;
• calculate missing angles using triangle, polygon, parallel-line and vertically opposite angle properties;
• apply Pythagoras’ theorem to find a hypotenuse or shorter side;
• use the converse of Pythagoras’ theorem to test whether a triangle is right-angled;
• solve right-angled triangle problems using sine, cosine and tangent;
• calculate unknown angles using ${\sin }^{-1}$, ${\cos }^{-1}$