Geometric Bites

New Mathematics Resources More Ways to Explore, Search and Study Mathematics Q.E.D. Engine and Function Atlas are now live, joining Geometric Bites and the growing Geometry Insights archive as independent mathematics resources for students, teachers and anyone who enjoys mathematics. Mathematics is easier to study when the right material is easy to find. These projects approach that problem in different ways: searchable proofs, graphing resources, free articles and a growing premium archive of detailed mathematical work. Mathematics • Proof • Exploration Different ways to see the same mathematics. Search the proof. Explore the graph. Read the mathematics. From classical results to modern mathematical resources — these...

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...

Fractions | Algebra | First Principles The familiar rules for adding and subtracting fractions can be derived directly from the properties of real numbers. The key idea is that multiplying a number by 1 does not change its value. By writing 1 in a suitable fractional form, both fractions can be expressed with the same denominator. Assumptions Let a, b, c, d ∈ ℝ, with b ≠ 0 and d ≠ 0. Addition of Two Fractions Begin with two fractions whose denominators are not necessarily equal: a b + c d = a b · 1 + c d · 1 = a b ...

Let A be a non-zero vector in ℝ³, and also regard its endpoint as the point through which the plane passes: A = (a 1 , a 2 , a 3 ) T ≠ (0, 0, 0) T . Let P be an arbitrary point on the plane: P = (x, y, z) T . The displacement from A to P is P − A = (x − a 1 , y − a 2 , z − a 3 ) T . Because the plane is perpendicular to A , the vector A is its normal vector. Every displacement P − A lying in the plane must therefore be perpendicular to A . Perpendicular vectors have a dot product of zero. Hence A · ( P − A ) = 0. Writing this condition in coordinates gives a 1 (x − a 1 ) + a 2 (y − a 2 ) + a 3 (z − a 3 ) = 0. Expanding the brackets: a 1 x − a 1 2 + a 2 y − a 2 2 + a 3 z − a 3 2 = 0. Move the squared terms to the right-hand side: a 1 x + a 2 y + a 3 z = a 1 2 + a 2 2 + a 3 2 . In vector notation, this becomes A · P = A · A = ‖ A ‖ 2 . ...

Let A, B and X be column vectors in ℝ³: A = a₁ a₂ a₃ , B = b₁ b₂ b₃ , X = x y z . BAᵀX = B(AᵀX). Matrix multiplication is associative, so AᵀX may be evaluated first. Since Aᵀ is a 1 × 3 row matrix and X is a 3 × 1 column matrix, their product is a scalar: AᵀX = a₁x + a₂y + a₃z. Therefore, BAᵀX = b₁ b₂ b₃ (a₁x + a₂y + a₃z). The quantity in parentheses is a scalar, so it multiplies every component of B: BAᵀX = b₁a₁x + b₁a₂y + b₁a₃z b₂a₁x + b₂a₂y + b₂a₃z b₃a₁x + b₃a₂y + b₃a₃z . Collect the coefficients of x, y and z into a matrix multiplying X: BAᵀX = b₁a₁ b₁a₂ b₁a₃ b₂a₁ b₂a₂ b...