Square Roots & Pythagorean Paths Math Worksheet (Age 13) — Common Core 8.G.B.6–8 & ACARA v9 — Beast Academy Ch.11

Instructions

Pray, attend to the following geometric predicaments. You are to employ the theorem attributed to the esteemed Pythagoras of Samos to discern the unknown quantities. A full and proper account of your reasoning is expected for each solution. You may leave answers that are not whole numbers in their square root form, as simplification to a mere decimal approximation is often a vulgarity.

Part I: The Propriety of Triangles

1. A surveyor claims that a triangular plot of land with sides measuring 9 metres, 12 metres, and 15 metres forms a perfect right-angled triangle. Do you find his assertion to be a matter of fact or a flight of fancy? Provide a mathematical proof for your conclusion.
2. Imagine a ladder, 20 feet in length, leaning against a vertical wall. The base of the ladder is positioned 12 feet from the base of the wall. To what height upon the wall does the top of the ladder ascend?

Part II: The Shortest Path

Consider the most direct route, for a straight line is the path of a mind clear of distraction.

3. Two friends, residing at different points upon a town map, wish to know the direct distance between their dwellings. Let us place the map upon a coordinate grid. The home of the first is situated at point P(-4, -2), and the home of the second at point Q(4, 4). What is the shortest distance, in units, between their two houses?
4. A small garden spider has spun its web upon a rectangular window pane that measures 8 inches in height and 15 inches in width. The spider sits at the bottom-left corner and spies a fly trapped at the top-right corner. What is the shortest distance the spider must crawl along the glass to reach its dinner?

Part III: A Three-Dimensional Conundrum

Herein lies a challenge requiring a greater measure of ingenuity. You must flatten a solid in your mind's eye to reveal the true path.

5. An ant, a creature of admirable industry, finds itself upon the floor at one corner of a wooden chest. The chest's dimensions are a length of 7 inches, a width of 4 inches, and a height of 2 inches. The ant wishes to travel to the interior corner on the lid diagonally opposite its starting point. What is the absolute shortest distance the ant can crawl along the surfaces of the chest to reach its destination? (Hint: The shortest path may not be the one that appears most obvious. Consider unfolding the chest in different ways.)

Part IV: A Spiral of Surds

One calculation builds upon the last, in a most pleasing and logical succession.

6. We shall begin with a humble right-angled triangle, whose two legs are each precisely 1 unit in length. Let us call its hypotenuse h₁. Now, construct a second right-angled triangle using h₁ as one leg and a new leg of length 1 unit. Let us call the hypotenuse of this second triangle h₂. Finally, construct a third right-angled triangle using h₂ as one leg and another new leg of length 1 unit. What, pray tell, is the length of the hypotenuse of this third triangle?

A Lady's & Gentleman's Guide to Scholarly Assessment

Being an Analytic Rubric for Judging Accomplishment in the Pythagorean Arts, as Aligned with the Australian Curriculum (ACARA v9) for the Eighth to Twelfth Year of Study

It is a truth universally acknowledged that a student in possession of a good mind must be in want of a proper assessment. The following rubrics, therefore, are presented for the discerning educator, to judge with propriety and fairness the intellectual efforts of a scholar in matters of geometry and measurement.

For a Scholar in their Eighth Year (AC9M8M06):

• A Profound Accomplishment (A): The student exhibits a mastery of the Theorem that is both elegant and complete, applying it with unerring accuracy to familiar and unfamiliar predicaments alike. Their reasoning is set down with such clarity as to leave no room for doubt.
• A Commendable Acquaintance (B): The student demonstrates a sound and reliable understanding, correctly solving problems of a standard nature. If they falter, it is only when faced with a novel arrangement or a multi-step conundrum, their method being sound, if not always perfectly executed.
• A Satisfactory Grasp (C): The student can apply the formula to find an unknown side of a right-angled triangle when the problem is presented in a straightforward manner. Their efforts may show some trifling errors in calculation or a degree of hesitation when the path to the solution is not immediately apparent.
• A Developing Understanding (D): The student possesses a nascent awareness of the relationship between the sides, but their application of the Theorem is marred by frequent error or misapprehension, suggesting a need for further, patient tutelage.

For a Scholar in their Ninth and Tenth Years (AC9M9M07, AC9M10M01):

• Of the First Rank of Excellence (A): The student moves with confidence from two-dimensional planes to the more complex society of three-dimensional solids. They can deconstruct a complex problem, such as that faced by our industrious ant, into a series of right-angled triangles, and solve it with ingenuity and precision, often discerning the most efficient path amongst several possibilities. Their work displays a true intellectual sensibility.
• A Display of Sound Judgement (B): The student capably navigates three-dimensional space, though they may require some gentle guidance or a hint to unfold the solid in the most advantageous manner. Their calculations are proper and their understanding of the underlying principles is not in question.
• An Adequate Performance (C): The student shows competence in two-dimensional applications, including those on a coordinate plane, but finds the transition to three dimensions to be a source of some bewilderment. Their attempts are earnest but may lack the spatial reasoning required for a correct solution to the more demanding problems.

For a Scholar of Advanced Years (Eleventh and Twelfth Year, Mathematical Methods & Specialist Mathematics):

• A Truly Distinguished Mind (A): The scholar perceives the Pythagorean Theorem not as an isolated rule, but as a foundational principle of Euclidean geometry, the very essence of the distance formula, and a concept readily expressed in the language of vectors. Their solutions are not merely correct, but abstract and general in their nature, demonstrating a profound connection between disparate mathematical realms.
• A Scholar of Considerable Merit (B): The scholar reliably applies the theorem in varied and complex contexts, including abstract coordinate and vector geometry. They may, however, articulate their generalised proofs with less polish or fail to grasp the most subtle of its theoretical implications, though their practical skill is beyond reproach.
• A Sufficient, if Uninspired, Grasp (C): The scholar's knowledge is sufficient for routine applications within their course of study, but they exhibit little inclination or ability to extend the principle to more abstract or generalised situations. Their understanding is functional rather than deep, and their performance is respectable, though it lacks brilliance.

Answer Key

1. The assertion is a matter of fact.

   By the converse of the Pythagorean theorem, if a² + b² = c², the triangle is a right-angled triangle. We test the sides:
   9² + 12² = 81 + 144 = 225
   15² = 225
   As 225 = 225, the plot of land is indeed a right-angled triangle.

2. The ladder ascends to a height of 16 feet.

   The ladder forms the hypotenuse. Let the height be h.
   h² + 12² = 20²
   h² + 144 = 400
   h² = 400 - 144
   h² = 256
   h = √256 = 16 feet.

3. The distance is 10 units.

   We form a right triangle where the legs are the change in the x-coordinates and the change in the y-coordinates.
   Change in x = 4 - (-4) = 8 units.
   Change in y = 4 - (-2) = 6 units.
   distance² = 8² + 6²
   distance² = 64 + 36 = 100
   distance = √100 = 10 units.

4. The shortest distance is 17 inches.

   The path is the hypotenuse of a right triangle with the window's width and height as its legs.
   distance² = 15² + 8²
   distance² = 225 + 64 = 289
   distance = √289 = 17 inches.

5. The shortest distance is √65 inches.

   We must "unfold" the chest to create a flat surface. There are three possibilities for the ant's path:
   1. Unfolding across the length and height: The new flat rectangle has legs of (7+2) and 4. Path² = 9² + 4² = 81 + 16 = 97.
   2. Unfolding across the width and height: The new flat rectangle has legs of (4+2) and 7. Path² = 6² + 7² = 36 + 49 = 85.
   3. Unfolding across the length and width (onto the floor/lid): The new flat rectangle has legs of (7+4) and 2. Path² = 11² + 2² = 121 + 4 = 125.
   Comparing the squared distances (97, 85, 125), the smallest is 85. Therefore, the shortest path is √85 inches.

6. The length of the third hypotenuse is 2 units.

   First triangle: Legs are 1 and 1. (h₁)² = 1² + 1² = 2. So, h₁ = √2.
   Second triangle: Legs are h₁ (√2) and 1. (h₂)² = (√2)² + 1² = 2 + 1 = 3. So, h₂ = √3.
   Third triangle: Legs are h₂ (√3) and 1. (h₃)² = (√3)² + 1² = 3 + 1 = 4. So, h₃ = √4 = 2 units.