Square Roots & Pythagorean Paths Worksheet (Age 15) — Beast Academy Ch.11 | ACARA v9 & Common Core

Free worksheet covering multiple subjects for age 15 students. Created with Learning Corner's AI-powered tools.

Get personalized worksheets for your own interests and needs

Try Worksheets Now
PDF
Worksheet

Instructions

Pray, attend to the following mathematical examinations. You are to employ the theorem credited to Pythagoras to discern the properties of the triangles and paths herein described. Present your solutions with all due diligence, ensuring that any square roots are expressed in their most simplified form. Let your reasoning be as clear as a country morning.

Part I: On the Nature of Triangles

For each set of given side lengths, determine if they might form a right-angled triangle. You must justify your conclusion with a calculation, as a mere assertion will not suffice.

  1. A triangle with sides of length 8 cm, 15 cm, and 17 cm.
  2. A triangle with sides of length 7 ft, 9 ft, and 12 ft.

Part II: In Search of a Missing Side

The following right-angled triangles are missing a dimension. Calculate the length of the unknown side, leaving your answer in the simplest radical form where necessary.

  1. The two shorter sides (legs) measure 5 inches and 6 inches. What is the measure of the hypotenuse?
  2. A right triangle possesses a hypotenuse of 14 metres and one leg of 8 metres. What is the length of the remaining leg?

Part III: A Practical Application

Consider a scenario of some consequence: A fine tapestry, measuring 8 feet in height and 12 feet in width, is to be adorned with a ribbon stretching directly from one corner to the diagonally opposite corner. What is the minimum length of ribbon required for this task? Provide your answer in simplest radical form.

Part IV: A Pythagorean Path

Imagine a lady undertaking a constitutional upon a grid representing a vast country estate. Her journey is composed of two distinct, straight paths.

First, she travels from the Manor House, situated at coordinate (1, 2), directly to the old Oak Tree at (9, 8).

Then, feeling reflective, she proceeds from the Oak Tree at (9, 8) to the quiet Bridge over the stream at (4, -4).

  1. What is the precise distance of the first leg of her journey, from the Manor to the Tree?
  2. What is the precise distance of the second leg of her journey, from the Tree to the Bridge?
  3. What is the total length of her journey? Express your answer as a sum of the two distances, each in its simplest form.

Answer Key

Part I: On the Nature of Triangles

  1. We test if a² + b² = c². The longest side, 17, must be the hypotenuse (c).

    8² + 15² = 64 + 225 = 289

    17² = 289

    As 289 = 289, it is indeed a right-angled triangle.

  2. We test if a² + b² = c². The longest side, 12, must be the hypotenuse (c).

    7² + 9² = 49 + 81 = 130

    12² = 144

    As 130 ≠ 144, it is not a right-angled triangle.

Part II: In Search of a Missing Side

  1. a² + b² = c²

    5² + 6² = c²

    25 + 36 = c²

    61 = c²

    c = √61 inches. (√61 cannot be simplified.)

  2. a² + b² = c²

    a² + 8² = 14²

    a² + 64 = 196

    a² = 196 - 64 = 132

    a = √132 = √(4 * 33) = 2√33 metres.

Part III: A Practical Application

The height and width form the legs of a right triangle, and the ribbon is the hypotenuse.

a² + b² = c²

8² + 12² = c²

64 + 144 = c²

208 = c²

c = √208 = √(16 * 13) = 4√13 feet.

Part IV: A Pythagorean Path

  1. Manor (1, 2) to Tree (9, 8).

    Change in x (Δx) = 9 - 1 = 8.

    Change in y (Δy) = 8 - 2 = 6.

    Distance² = 8² + 6² = 64 + 36 = 100.

    Distance = √100 = 10 units.

  2. Tree (9, 8) to Bridge (4, -4).

    Change in x (Δx) = 9 - 4 = 5.

    Change in y (Δy) = 8 - (-4) = 12.

    Distance² = 5² + 12² = 25 + 144 = 169.

    Distance = √169 = 13 units.

  3. Total journey length = (Distance of first leg) + (Distance of second leg).

    Total = 10 + 13 = 23 units.

A Tutor’s Analytical & Scoring Rubric

Regarding a Scholar's Command of the Pythagorean Theorem, with Reference to the Australian Curriculum (ACARA v9)

Level of Attainment Qualities and Characteristics Observed (Years 8-10) Prospects for Future Studies (Years 11-12)
A Most Accomplished Scholar The student demonstrates a command of the Pythagorean principle as thorough and elegant as a well-executed quadrille. They navigate not only the straightforward calculations (AC9M8M06) but also the more nuanced paths upon the Cartesian plane (AC9M10M01) with a pleasing alacrity and precision. Their reasoning is transparent, their simplification of radicals masterful, and their application to worldly problems shows a depth of understanding that speaks of a truly well-furnished mind. All workings are presented with admirable clarity. Such a profound and intuitive grasp of spatial reasoning predicts a most favourable outcome in the advanced studies of Mathematical Methods and Specialist Mathematics. One foresees no impediment to their comprehending vectors in two and three dimensions, nor the geometric proofs that shall be required of them.
A Sound and Sensible Understanding The scholar exhibits a creditable and correct application of the theorem. They can, with respectable consistency, determine the length of a missing side and verify the nature of a triangle (AC9M8M06). When faced with a problem of application or a path upon a grid (AC9M9M01), their approach is logical, though perhaps lacking the swift elegance of the most accomplished. Minor imperfections may appear in the simplification of a surd, but the fundamental principles are firmly in their possession. This student possesses a character of sound mathematical sense, fit for the continued study of General Mathematics. With diligence, they may find success in Mathematical Methods, provided they attend to the finer points of algebraic manipulation and multistep reasoning, which at present are merely adequate rather than exemplary.
A Promising, if Hasty, Endeavour Here we observe a mind that has grasped the general idea, yet is prone to misadventure in the execution. The formula is known, but may be misapplied—confusing a leg for a hypotenuse, for instance. There is a discernible effort to find distances on the plane (AC9M10M01), but calculations are marred by errors of arithmetic or a failure to correctly simplify radicals. The work displays potential, but is wanting in the discipline of careful review. Further progress is contingent upon the cultivation of greater precision and attention to detail. Without such improvement, the abstract nature of senior mathematics will prove a considerable challenge. Remedial exercises and a more methodical manner are earnestly recommended before embarking upon such a course.
Requiring Further Tutelage The scholar's acquaintance with the Pythagorean theorem is, at present, slight and uncertain. There is significant confusion regarding the relationship between the sides of a right-angled triangle, and attempts to apply the formula are frequently incorrect from their very inception. The foundational concepts (AC9M8M06) appear not yet to have taken root, and consequently, any application thereof is built upon precarious ground. It is of the utmost importance that these fundamental deficits be addressed with haste. To proceed to the higher branches of mathematics without a firm understanding of these principles would be an act of profound folly, akin to building the upper storeys of a house with no foundation. Immediate and focused instruction is required.
With Worksheets, you can:
  • Reinforce key concepts
  • Provide hands-on practice
  • Customize exercises to fit your needs
  • Track your student's improvement
Try Worksheets Now