Instructions
You find before you a series of challenges concerning right-angled triangles and the remarkable theorem attributed to Pythagoras. Consider each problem with care. You are encouraged to draw diagrams to aid your reasoning. Present your solutions with clarity, ensuring your method is as elegant as your final answer. A calculator may be of service for arithmetic, but the true work lies in your logic.
Part 1: Finding the Unknown
For the right-angled triangles described below, determine the length of the unknown side, denoted by x. Provide your answer in the simplest radical form where necessary.
- The two legs have lengths of 7 cm and 24 cm. Find the length of the hypotenuse, x.
- A right triangle has a leg of length 9 m and a hypotenuse of length 15 m. Find the length of the other leg, x.
- The legs of a right triangle have lengths of 5 inches and 10 inches. What is the length of the hypotenuse, x?
Part 2: A Practical Dilemma
A rectangular garden measures 16 feet in width and 30 feet in length. You wish to run a string diagonally from one corner to the opposite corner to plan a new path. What is the shortest possible length of the string required to span the diagonal of the garden?
Part 3: The Isosceles Challenge
Consider an isosceles triangle with two sides of length 13 cm and a base of length 10 cm. By drawing a line from the top vertex perpendicular to the base, you can create two congruent right-angled triangles. Use this knowledge and the Pythagorean theorem to calculate the area of the isosceles triangle.
Part 4: Right or Not Right?
The converse of the Pythagorean theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides (if a² + b² = c²), then the triangle is a right-angled triangle. Determine if a triangle with the following side lengths is a right triangle. Justify your answer with calculations.
Side lengths: 10, 24, and 26.
Answer Key
Part 1: Finding the Unknown
-
Solution: Let the legs be a = 7 and b = 24. The hypotenuse is c = x.
a² + b² = c²
7² + 24² = x²
49 + 576 = x²
625 = x²
x = √625
x = 25 cm -
Solution: Let the known leg be a = 9 and the hypotenuse be c = 15. The unknown leg is b = x.
a² + b² = c²
9² + x² = 15²
81 + x² = 225
x² = 225 - 81
x² = 144
x = 12 m -
Solution: Let the legs be a = 5 and b = 10. The hypotenuse is c = x.
5² + 10² = x²
25 + 100 = x²
125 = x²
x = √125
x = √(25 * 5)
x = 5√5 inches
Part 2: A Practical Dilemma
Solution: The garden's width and length form the legs of a right triangle, and the diagonal is the hypotenuse.
Let a = 16 and b = 30. We must find c.
16² + 30² = c²
256 + 900 = c²
1156 = c²
c = √1156
c = 34 feet
The shortest possible length of the string is 34 feet.
Part 3: The Isosceles Challenge
Solution: The perpendicular line (the height, h) bisects the base, creating two right triangles. Each right triangle has a hypotenuse of 13 cm and one leg of 5 cm (half of the 10 cm base). The height h is the other leg.
h² + 5² = 13²
h² + 25 = 169
h² = 144
h = 12 cm
Now, we find the area of the original isosceles triangle: Area = (1/2) * base * height.
Area = (1/2) * 10 cm * 12 cm
Area = 60 cm²
Part 4: Right or Not Right?
Solution: To check, we must see if the sum of the squares of the two shorter sides equals the square of the longest side.
Let a = 10, b = 24, and c = 26.
Does a² + b² = c²?
Does 10² + 24² = 26²?
100 + 576 = 676
676 = 676
Yes, the equality holds true. Therefore, a triangle with side lengths of 10, 24, and 26 is indeed a right triangle.
A Most Judicious Assessment of a Scholar's Mathematical Accomplishments
Analytic Scoring Rubric for the Years Eight through Twelve, Aligned with the Principles of the ACARA v9
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. Henceforth, the following criteria shall serve to evaluate the merits of a student’s work, that their progress may be most properly ascertained.
| Criterion of Judgement | A Most Accomplished Scholar (Exceeding) | A Commendable Understanding (Proficient) | A Promising Endeavour (Developing) | Requires Further Guidance (Beginning) |
|---|---|---|---|---|
| Application of Theorems and Formulae AC9M8M06: Solves problems involving the Pythagorean theorem. |
The student applies the Pythagorean theorem with a decided felicity, not only to familiar arrangements but to novel and complex situations, demonstrating an intuition for geometric propriety. | The theorem is employed with correctness and propriety in all customary circumstances. The calculations are executed with accuracy, revealing a sound and well-tutored comprehension. | The student shows a burgeoning acquaintance with the theorem, applying it correctly to straightforward figures, yet may exhibit some diffidence or error when faced with a more intricate dilemma. | The student’s grasp of the theorem appears tenuous. There is a marked inconsistency in its application, and one observes a general confusion between the legs and the hypotenuse of the affair. |
| Mathematical Reasoning and Communication AC9M9A01: Communicates mathematical reasoning and arguments using formal language. |
The scholar's reasoning is conveyed with such elegance and clarity as to be irrefutable. Each step follows the last with logical grace, forming an argument of admirable strength and economy. | The solution is presented in a sensible and orderly fashion. The reasoning is quite clear, and the method of calculation is made plain to the observer, as any respectable account ought to be. | An attempt is made to show the progression of thought, yet the account may be wanting in certain particulars or suffer from a want of logical connection, leaving the observer to surmise the intermediate steps. | The final declaration, or answer, is presented with little to no supporting evidence. The path to the conclusion is shrouded in such obscurity that its validity cannot be properly judged. |
| Problem Solving and Modelling AC9M10P02: Formulates a mathematical model from a real-world context. |
With perceptive insight, the student discerns the geometric heart of a narrative problem, translating worldly scenarios into mathematical models with ingenuity and strategic foresight. | When presented with a problem of a practical nature, the student correctly identifies the necessary geometric relationships and constructs a suitable model for its resolution. | The student can perceive the outline of a mathematical model within a given context but may struggle to define all variables and relationships with the required precision. | The student finds it a great trial to abstract a mathematical structure from a described situation, and the resulting model, if any, bears little resemblance to the circumstances at hand. |