Instructions
Pray, attend to the following mathematical investigations. You are to employ the principles of right-angled triangles as established by the esteemed Pythagoras. Approach each problem with careful thought and demonstrate your reasoning with clarity. You may leave your answers in square root form where appropriate, lest the numbers become unwieldy.
Part 1: The Foundations of Right-Angled Figures
The relationship in any right-angled triangle is given by the formula a2 + b2 = c2, where a and b are the lengths of the two shorter sides (the legs) and c is the length of the longest side (the hypotenuse).
- A triangle possesses legs of length 8 cm and 15 cm. What, then, is the length of its hypotenuse?
- A right-angled triangle has a hypotenuse of 25 inches and one leg of 7 inches. Ascertain the length of the remaining leg.
- A ladder, 13 feet in length, is leaned against a vertical wall. The base of the ladder is positioned 5 feet away from the base of the wall. To what height upon the wall does the ladder extend?
Part 2: Navigating the Cartesian Plane
One may employ the Pythagorean theorem to determine the most direct distance between two points on a coordinate grid. Imagine the straight line between the points as the hypotenuse of a right-angled triangle, with the legs formed by the horizontal and vertical distances.
- Consider the two points, Point A at (2, 1) and Point B at (7, 8), plotted upon a grid. What is the precise, straight-line distance between them?
- A diligent spider is situated at corner P of a large, empty rectangular room that measures 12 metres in length and 9 metres in width. A most unfortunate fly has become trapped at the diagonally opposite corner, Q. The spider, being unable to fly, must traverse the floor. What is the shortest possible distance the spider can walk to reach the fly?
- Imagine a journey upon a grid. You begin at the origin (0,0). Your path consists of the following movements: 6 units east (positive x-direction), followed by 4 units north (positive y-direction), followed by 3 units west (negative x-direction), and finally 4 units north again. What is the straight-line distance from your starting point to your final position?
Part 3: Advanced Applications & Elegant Structures
The principles we have examined may be extended to more complex figures and situations, revealing the underlying structure of geometry.
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Pythagorean Triples. Certain right-angled triangles are possessed of sides whose lengths are all whole numbers. A well-known family of these is the {3, 4, 5} triangle and its multiples {6, 8, 10}, {9, 12, 15}, etc. Another such family is {5, 12, 13}.
A ship sails 50 leagues due west from port. It then turns and sails 120 leagues due south. What is the direct distance from the ship back to the port?
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Rearranging Formulae. Consider a square with side length s. The diagonal, d, divides the square into two identical 45-45-90 triangles.
a) Write a formula for the length of the diagonal, d, in terms of the side length, s. (That is, d = ...)
b) Now, rearrange this formula to express the side length, s, in terms of the diagonal, d. (That is, s = ...) -
Three-Dimensional Space. A rectangular box has a length of 12 cm, a width of 4 cm, and a height of 3 cm. What is the length of the longest straight line that can be drawn from one corner of the box to the diagonally opposite corner? (This is known as the space diagonal).
Hint: You may need to apply the theorem twice. First, find the diagonal of the base of the box.
Answer Key
- a2 + b2 = c2 => 82 + 152 = c2 => 64 + 225 = c2 => 289 = c2. Thus, c = 17 cm.
- a2 + b2 = c2 => 72 + b2 = 252 => 49 + b2 = 625 => b2 = 625 - 49 = 576. Thus, b = 24 inches.
- The ladder is the hypotenuse. a2 + 52 = 132 => a2 + 25 = 169 => a2 = 144. Thus, the height a = 12 feet.
- The horizontal distance (change in x) is 7 - 2 = 5 units. The vertical distance (change in y) is 8 - 1 = 7 units. These are the legs of the triangle. 52 + 72 = c2 => 25 + 49 = c2 => c2 = 74. The distance is √74 units.
- The path across the floor is the hypotenuse of a right triangle with legs equal to the length and width of the room. 122 + 92 = c2 => 144 + 81 = c2 => 225 = c2. The distance is 15 metres.
- Final x-position: 6 (east) - 3 (west) = 3. Final y-position: 4 (north) + 4 (north) = 8. The final coordinates are (3, 8). The distance from the origin (0,0) is the hypotenuse of a triangle with legs of 3 and 8. 32 + 82 = c2 => 9 + 64 = c2 => c2 = 73. The distance is √73 units.
- This problem forms a right-angled triangle with legs of 50 and 120 leagues. One might notice this is a multiple of the {5, 12, 13} triple, scaled by 10. The hypotenuse must therefore be 13 * 10 = 130. Calculation: 502 + 1202 = c2 => 2500 + 14400 = c2 => 16900 = c2. The distance is 130 leagues.
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a) The legs are both s and the diagonal is d. So, s2 + s2 = d2 => 2s2 = d2. To solve for d, we take the square root: d = s√2.
b) Starting from 2s2 = d2, we solve for s. Divide by 2: s2 = d2/2. Take the square root: s = d/√2 (or s = (d√2)/2). - First, find the diagonal of the 12 cm by 4 cm base (let's call it db). 122 + 42 = db2 => 144 + 16 = db2 => db2 = 160. Now, this base diagonal (√160) and the height (3 cm) form the legs of a new right triangle, whose hypotenuse is the space diagonal (let's call it ds). (db)2 + 32 = ds2 => 160 + 9 = ds2 => 169 = ds2. The space diagonal is 13 cm. (A quicker method: d2 = l2 + w2 + h2 = 122 + 42 + 32 = 144 + 16 + 9 = 169, so d = 13).
A Teacher's Analytical Rubric for the Assessment of a Scholar's Progress in the Geometric Arts
In accordance with the Australian Curriculum, Version 9.0 (Years 8-12)
| Criterion of Judgement | A Scholar of Great Perspicacity (Exceeding) |
A Most Commendable Acquaintance (Achieved) |
A Promising, Albeit Hesitant, Endeavour (Developing) |
|---|---|---|---|
Application of Mr. Pythagoras's Theorem(AC9M8A01) |
The scholar executes calculations with an unimpeachable precision and swiftness, finding both hypotenuse and leg with equal and unerring facility. Their work displays an innate sense of propriety for the theorem's application. | The student demonstrates a sound and proper grasp of the theorem's primary function, correctly calculating unknown sides in straightforward constructions. Their method is clear and their conclusions, for the most part, correct. | There is evidence of a mind striving for comprehension; however, the scholar is apt to make trifling errors in computation or may misapply the formula, perhaps confounding the hypotenuse with a leg of the figure. |
Spatial Reasoning upon the Cartesian Plane(AC9M8SP03, AC9MFM09) |
With remarkable deftness, the student translates coordinates into the legs of a right-angled triangle. They navigate complex paths and three-dimensional space with a logical grace that speaks to a true and deep understanding. | The scholar capably determines the distance between two points on a plane and can solve simple word problems involving spatial contexts. The connection between the grid and the theorem is well understood. | The student can plot points correctly but may falter when required to construct the abstract triangle needed for calculation. Multi-step navigation problems prove a considerable challenge to their present sensibilities. |
Algebraic Dexterity and Formulation(AC9M10A02, A-SSE.1, A-CED.4) |
It is with great satisfaction that one observes the scholar's ability to perceive underlying algebraic structures, such as Pythagorean triples. They rearrange formulae with elegance and express relationships in a generalised, abstract manner. | The student is able to follow a guided path towards rearranging a formula and can recognise a simple integer triple when it is presented. They can substitute values into a given formula correctly. | The manipulation of variables proves to be a source of confusion. The scholar prefers the comfortable certainty of numbers and struggles to work with formulae in their more general, and hence more powerful, form. |
Logical Exposition and Problem Dissection(All strands) |
The scholar's reasoning is a model of clarity. They dissect multifaceted problems into their constituent parts and present a solution that is not merely correct, but logically and aesthetically satisfying. Their workings are presented with utmost neatness and order. | The student presents their work in a coherent fashion, showing the necessary steps to arrive at a solution. They are able to extract the relevant mathematical information from a moderately complex narrative. | The path to a solution is often obscured by a disorganised presentation or a failure to identify the central question within a word problem. A more methodical and orderly approach is earnestly recommended for future endeavours. |