Instructions
You are a cartographer's apprentice, tasked with mapping treacherous paths and mysterious structures. The Pythagorean Theorem shall be your most trusted instrument. Use it to determine the true lengths of the paths described below. Show your calculations with due diligence. Where necessary, answers should be rounded to two decimal places.
Part I: The Shipwright's Dilemma
1. A ship's mast and the deck form a perfect right angle. A support rope measures 17 metres in length and is fastened to the deck at a point 8 metres from the base of the mast. What is the height of the mast where the rope is attached?
2. A small boat requires a new sail in the shape of a right triangle. Its two shortest sides, which form the right angle, are to be 9 feet and 12 feet long. What must the length of the longest side (the hypotenuse) be?
Part II: A Question of Righteousness
3. A lord wishes to build a triangular garden plot. He has instructed his groundskeeper to create a design with sides measuring 10 yards, 12 yards, and 15 yards. Can this garden possibly contain a perfect 90-degree corner? Explain your reasoning using a calculation.
Part III: Charting the Estate
4. On your map of the estate, the Manor House is located at coordinate M(2, 3) and the Old Mill is found at O(7, 15). What is the direct, straight-line distance between these two locations? You may find it helpful to sketch a right triangle using the grid lines as the legs.
Part IV: The Secret of the Tower
5. An architect has designed a grand, rectangular hall that is 20 metres long, 16 metres wide, and 12 metres high. A clever spider, wishing to travel from one bottom corner of the hall to the very opposite top corner, seeks the shortest possible path. What is the length of this direct, straight-line journey through the air inside the hall?
Part V: The Pilgrim's Path
6. A pilgrim's journey from the Abbey to the Beacon consists of three straight segments. Using the map coordinates below, calculate the length of each segment and then find the total distance of the pilgrim's path.
- Path 1: From the Abbey at A(-5, 1) to the Bridge at B(-1, 4)
- Path 2: From the Bridge at B(-1, 4) to the Shrine at S(5, 4)
- Path 3: From the Shrine at S(5, 4) to the Beacon at K(8, 0)
What is the total length of the journey?
Answer Key
1. The Shipwright's Dilemma (1):
Let the mast height be b. Using a² + b² = c², we have 8² + b² = 17².
64 + b² = 289
b² = 289 - 64 = 225
b = √225 = 15.
Answer: 15 metres.
2. The Shipwright's Dilemma (2):
Let the hypotenuse be c. Using a² + b² = c², we have 9² + 12² = c².
81 + 144 = c²
c² = 225
c = √225 = 15.
Answer: 15 feet.
3. A Question of Righteousness:
To be a right triangle, the sides must satisfy the Pythagorean theorem. The longest side must be the hypotenuse. Let us check if 10² + 12² = 15².
100 + 144 = 244.
15² = 225.
Since 244 ≠ 225, the converse of the Pythagorean theorem tells us it is not a right triangle.
Answer: No, the garden cannot have a 90-degree corner because the side lengths do not satisfy a² + b² = c².
4. Charting the Estate:
The horizontal distance (Δx) is 7 - 2 = 5.
The vertical distance (Δy) is 15 - 3 = 12.
These are the legs of the right triangle. The distance (d) is the hypotenuse.
5² + 12² = d²
25 + 144 = 169
d = √169 = 13.
Answer: 13 units.
5. The Secret of the Tower:
This requires two steps. First, find the diagonal of the floor (d_floor).
d_floor² = 20² + 16² = 400 + 256 = 656.
Now, use the floor diagonal and the hall's height as the legs of a new right triangle to find the space diagonal (D).
D² = (d_floor)² + height² = 656 + 12² = 656 + 144 = 800.
D = √800 ≈ 28.28.
Answer: Approximately 28.28 metres.
6. The Pilgrim's Path:
- Path 1 (A to B): Δx = -1 - (-5) = 4. Δy = 4 - 1 = 3. Length = √(4² + 3²) = √(16 + 9) = √25 = 5 units.
- Path 2 (B to S): This is a horizontal line. Δx = 5 - (-1) = 6. Δy = 4 - 4 = 0. Length = 6 units.
- Path 3 (S to K): Δx = 8 - 5 = 3. Δy = 0 - 4 = -4. Length = √(3² + (-4)²) = √(9 + 16) = √25 = 5 units.
For the Esteemed Educator:
On the Assessment of a Student’s Mathematical Accomplishments
in the Application of the Pythagorean Theorem
It is a truth universally acknowledged that a young mind in possession of a good fortune of knowledge must be in want of a proper assessment. The following rubrics are offered as a guide to discerning the true character of a student’s understanding, aligned with the most proper curricula of the Australian realm (ACARA Version 9; Years 8–12).
| Criterion | A Student of Accomplished Understanding (A) | A Student of Commendable Sense (B) | A Student of Developing Judgement (C) | A Student in Want of Instruction (D/E) |
|---|---|---|---|---|
| Discernment of the Pythagorean Principle AC9M8M06 |
The student, with laudable sagacity, correctly identifies all circumstances wherein the theorem is of use, including its converse, and proceeds without hesitation or error. | The student generally applies the theorem correctly but may exhibit a moment's confusion when faced with its converse or a less common arrangement (e.g., solving for a leg). | The student can employ the theorem for a simple hypotenuse but is frequently confounded when a leg is to be found or when a judgement upon a non-right triangle is required. | The student’s attempts are but a matter of conjecture, applying the formula with little regard for its true meaning, often confusing legs and hypotenuse. |
| Judicious Application in Two Dimensions AC9M9M01 |
The student navigates the Cartesian plane with the elegance of a dancer at a ball, deriving distances from coordinates with a fastidious precision and unerring accuracy. | The student is capable of finding the path between two points, though their calculations may lack a certain elegance and are occasionally marred by a minor arithmetic misstep. | The student requires the aid of a pre-drawn triangle upon the grid to make any headway and struggles to translate coordinates into the requisite lengths independently. | The student appears quite lost on the plane, as if in a hedge maze, unable to determine the horizontal and vertical separations that form the basis of the calculation. |
| Facility with Three-Dimensional Space AC9M9M05, AC9M10M01 |
The student exhibits a remarkable capacity for spatial reasoning, extending the theorem into the third dimension with confidence and resolving the longest diagonal as if it were a trifling parlour game. | The student possesses the general notion of the method, but the multi-step process may prove a slight tax upon their faculties, leading to errors in the intermediate calculations. | The student attempts to apply the theorem in three dimensions but does so incorrectly, often by merely summing the squares of all three dimensions at once, a grievous but common error. | The very notion of a third dimension appears to be a source of great perplexity, and no reasonable attempt is made to solve the problem. |
| Clarity and Precision of Argument (General Proficiency) |
The student’s workings are a model of propriety and order. Each step follows logically from the last, and the final answer is presented with the appropriate degree of precision, a testament to a well-ordered mind. | The student’s reasoning is generally clear, though it may at times be cluttered with extraneous markings or want for a concluding statement or proper units. | The path to the student’s solution is a tangled one, difficult for an observer to follow, and the final answer is often presented without the proper units or rounding. | The workings are of such a chaotic and inscrutable nature that one cannot, with any certainty, discern the student's intent. |