Geometric Bites

Scale drawings are useful for representing large real-life objects and areas using smaller measurements. When calculating an area from a scale drawing, the scale factor must be squared because area is two-dimensional. Question – Area of a Park from a Scale Drawing A map has a scale of 1 : 40 000 . A rectangular park is shown on the map as measuring 5.8 cm by 4.4 cm . Calculate the area of the park in real life, giving the answer in km² to 3 significant figures . Working 1 : 4 × 10⁴ 5.8 : 5.8 × 4 × 10⁴ 4.4 : 4.4 × 4 × 10⁴ Area of park in real life: 5.8 × 4.4 × 4² × (10⁴)² cm² = 408.32 × 10⁸ cm² 1 m = 100 cm = 10² cm 1000 m = 1 km = 10⁵ cm 1 cm = 10⁻⁵ km (1 cm)² = (10⁻⁵ km)² 1 cm² = 10⁻...

These two examples use proportional relationships involving square roots. In each case, the proportional statement is first converted into an equation containing a constant of proportionality, k . The given values are then used to find k before the required value is calculated. Question 1 – Direct Proportion and Square Roots The time t seconds taken for a stone to fall is directly proportional to the square root of the distance d metres. When t = 4.6 , d = 25 . (a) Express t in terms of d . (b) Find the time taken when d = 42.25 . (a) Express t in terms of d t ∝ √d ⇒ t = k√d when t = 4.6, d = 25 ⇒ 4.6 = k√25 ⇒ 4.6 = 5k ⇒ k = 4.6 / 5 ∴ k = 0.92 t = 0.92√d ...

Density describes how much mass is contained within a particular volume. The relationship between density, mass and volume can be expressed using one formula, which can then be rearranged depending on the quantity that needs to be found. Density Density is calculated by dividing mass by volume: Density = mass / volume d = m / v What the Symbols Mean d = density m = mass v = volume Finding Volume Start with the density formula: d = m / v ⇒ dv = m v = m / d Volume = mass / density Finding Mass Starting again with: d = m / v ⇒ dv = m ...

Kinematics uses differentiation to connect displacement, velocity and acceleration. In this example, a stone is projected vertically upwards and its displacement is given as a function of time. Differentiation allows us to find its velocity and determine when it reaches its maximum height. The Displacement Function A stone is projected vertically upwards from the ground. After t seconds, its height above the ground, s metres, is given by: s(t) = 15t − 4.9t² 0 ≤ t ≤ 4 Question (a) – Find ds/dt Differentiate the displacement function with respect to time. Working s(t) = 15t − 4.9t² ds/dt = 15 − 4.9 × 2 × t = 15 − 9.8t Answer: ds/dt = 15 − 9.8t Question (b) – Velocity at...

These examples look at transformations of the graph y = f(x) . The original curve has a minimum point at (2, −1) . By examining how the function changes, we can determine how the coordinates of this minimum point are transformed. Original Minimum Point The curve y = f(x) has a minimum point at: (2, −1) Question (a)(i) – y = f(x + 2) Find the coordinates of the minimum point after the transformation y = f(x + 2) . Working y = f(x + 2) * Translation (−2, 0) Answer: (0, −1) Answer: (0, −1) Question (a)(ii) – y = 3f(x) Find the coordinates of the minimum point after the transformation y = 3f(x) . Working y = 3f(...

These examples practise converting units of area and volume, using powers of 10, writing values in standard form, and applying the pressure formula. Particular care is needed because area conversions are squared and volume conversions are cubed. Question 1(a) – Converting m² to cm² Convert 2.3 m² into cm² . Working Convert 2.3 m² into cm². 1 m = 10² cm 2.3 × (10² cm)² = 2.3 × 10⁴ cm² Answer: 2.3 × 10⁴ cm² Question 1(b) – Converting mm³ to cm³ Convert 400 mm³ into cm³ . Working Convert 400 mm³ into cm³ 10 mm = 1 cm 1 mm = 10⁻¹ cm 400 × (10⁻¹ cm)³ = 400 × 10⁻³ cm³ = 4 × 10² × 10⁻³ cm³ = 4 × 10⁻¹ cm³ Answer: 4 ×...

Pressure, force and area are connected by a simple formula. Once the relationship is understood, the formula can be rearranged depending on which quantity needs to be found. Pressure, Force and Area The basic relationship is: To find pressure: P = F / A To find force: F = P × A To find area: A = F / P What the Symbols Mean P represents pressure. F represents force. A represents area. Units * Pressure is measured in pascals (Pa), where 1 Pa = 1 N/m². * Force is measured in newtons (N). * Area (A) is measured in square metres (m²). Understanding the Formula Pressure describes h...

These ratio problems are based on questions from the Revise Pearson Edexcel International GCSE (9–1) Mathematics A – Higher Tier Revision Guide . The solutions below use an algebraic ratio method in which each part of the ratio is represented as a multiple of x . Question 1 – Sharing Money in a Ratio Andre, Becky and Makito share money in the ratio 3 : 6 : 7 . Andre and Becky receive £207 altogether. Work out how much Makito receives. Working A : B : M = 3x : 6x : 7x 3x + 6x = 207 9x = 207 x = 207/9 7 × (207/9) = 161 Makito receives £161. Answer: £161 Question 2(a) – Ages in a Ratio Amir and Petra's ages are in the ratio 3 : 7 . Amir is 9 years old . Work out Petra's age. Wo...

The associative, commutative and distributive laws are three of the most important structural rules in algebra. They explain how expressions may be grouped, reordered, expanded and simplified without changing their mathematical value. These laws are used throughout arithmetic, algebra, factorisation, equation solving and mathematical proof. Associative Law The associative law describes how terms may be grouped when the same operation is repeated. If an operation is associative, changing the placement of the brackets does not change the final value of the expression. For addition: a + (b + c) = (a + b) + c For example: 1 + (2 + 3) = (1 + 2) + 3 The associative law also applies to multiplication: a × (b × c) = (a × b) × c For example: 2 × (3 × 4) = (2 × 3) × 4 Subtraction is not associative because changing the grouping can change the result. a − (b − c) ≠ (a − b) − c For example: 1 − (2 − 3) ≠ (1 − 2) − 3 Commutative Law The commutative law describe...

These workings derive the laws of base-10 logarithms from exponent laws by converting between exponential form and logarithmic form. Assume a > 0 , b > 0 , and n ≠ 0 . Product Rule log(ab) = log a + log b Let 10 x = a Let 10 y = b Therefore: log 10 a = x log 10 b = y 10 x 10 y = ab 10 x+y = ab Therefore: log 10 (ab) = x + y Substituting: log 10 (ab) = log 10 a + log 10 b Therefore: log(ab) = log a + log b Quotient Rule log(a / b) = log a - log b Let 10 x = a Let 10 y = b Therefore: log 10 a = x log 10 b = y 10 x / 10 y = a / b 10 x-y = a / b Therefore: log 10 (a / b) = x - y Substituting: log 10 (a / b) = log 10 a - log 10 b Therefore: log(a / b) = log a - log b Power Rule log(a n ) = n log a Let log(a n ) = x Therefore: 10 x = a n Taking the n-th root of both sides: (10 x ) 1/n = (a n ) 1/n 10 x/n = a Therefore: log 10 a = x / n n log 10 a = x Theref...

A quadratic function and its turning point. Link to graph:  https://www.desmos.com/calculator/fktyfs12st A quadratic function is any function of the form f(x) = ax² + bx + c with a ≠ 0 . Its graph is a parabola, and every parabola has exactly one turning point (also called the vertex ). Completing the square is fundamental because it rewrites the quadratic as a shifted square , which makes the turning point immediately visible. Vertex form: the turning point is built in The vertex form of a quadratic is: f(x) = a(x − h)² + k This form is powerful because it exposes two facts at once: (x − h)² ≥ 0 for all real x (a square is never negative). (x − h)² = 0 happens exactly when x = h . So: If a > 0 , then a(x − h)² ≥ 0 , so the smallest possible value of f(x) is k , achieved at x = h (a minimum). If a < 0 , then a(x − h)² ≤ 0 , so the largest possible value of f(x) is k , achieved at x = h (a maximum). Therefore, in vertex form, the turning ...

Parabolas in sight: The Clifton Suspension Bridge, Bristol, United Kingdom. A quadratic function is a function whose graph is a parabola (a U-shaped curve). One of the most useful ways to write a quadratic is in vertex form , because it shows the parabola’s turning point immediately. 1) The vertex form A quadratic function in vertex form is written as: f(x) = a(x - h) 2 + k This form is especially helpful because the values h and k tell you the vertex directly. 2) The vertex (turning point) The vertex is the point where the parabola changes direction. In vertex form: Vertex = (h, k) If the parabola opens up , the vertex is the lowest point (a minimum). If the parabola opens down , the vertex is the highest point (a maximum). 3) What the number a does The number a controls two key things: the direction the parabola opens, and how wide or narrow it is. a > 0 means the parabola opens up (U-shape). a < 0 means the parabola opens dow...

Geometric Bites has always been about clear diagrams and full derivations that help you see the structure of mathematics. Over time it has grown into a rich, freely accessible library of explanations, proofs and visual ideas. Geometry Insights is the premium companion to that work: a dedicated article site focused on GCSE, A-Level and Further Pure mathematics, written from first principles with carefully engineered diagrams and a long-term, structured archive in mind. Visit Geometry Insights: https://geometryinsights.wordpress.com What Makes Geometry Insights Different? Where Geometric Bites offers free posts and full derivations, Geometry Insights is a curated, subscription-only library. Each article is built to answer a deeper question: not just “how do I use this formula?” but “why does this formula exist at all?” Premium-only articles that go in depth on GCSE, A-Level and Further Pure topics. First-principles derivations that start from definitions and basic f...

This article presents the rules of logarithms using complete, line-by-line derivations. Every identity is built directly from its exponential origin, without shortcuts, matching the structure of formal handwritten algebra. 1. Definition We begin with fundamental exponent facts: a⁰ = 1 ⇒ logₐ(1) = 0 a¹ = a ⇒ logₐ(a) = 1 Say: aᵐ = p Then, by definition: logₐ(p) = m Raise both sides of aᵐ = p to the power 1/m (with m ≠ 0 ): p^(1/m) = a Therefore: logₚ(a) = 1/m Since m = logₐ(p) , we obtain: logₐ(p) = 1 / logₚ(a) 2. Product Rule — Full Derivation Say: aᵐ = p and aⁿ = q Multiply: aᵐ · aⁿ = p · q Using index addition: a^(m+n) = p · q Taking logarithms: logₐ(p · q) = m + n Substitute: logₐ(p · q) = logₐ(p) + logₐ(q) 3. Quotient Rule — Full Derivation Say: aᵐ = p and aⁿ = q Divide: aᵐ / aⁿ = p / q Index subtraction gives: a^(m−n) = p / q Taking logarithms: logₐ(p / q) = m − n So: log...