Pythagoras Theorem

Core Concepts

The Pythagoras Theorem is a fundamental principle in geometry that relates the three sides of a right-angled triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side, which is opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

If a right-angled triangle has legs of lengths $a$ and $b$, and a hypotenuse of length $c$, the theorem is expressed mathematically as: $$a^2 + b^2 = c^2$$

Proof of Pythagoras Theorem

There are numerous ways to prove the Pythagoras theorem. A classic and intuitive algebraic proof involves the concept of areas.

Consider a square constructed with sides of length $a+b$. Inside this large square, we can arrange four identical right-angled triangles (each with base $a$, height $b$, and hypotenuse $c$). The arrangement leaves an empty smaller square in the center with side length $c$.

1. The area of the large outer square is given by: $(a+b)^2$.
2. Expanding this area yields: $a^2 + 2ab + b^2$.
3. The total area can also be found by adding the area of the four identical triangles and the area of the inner square. The area of one triangle is $\frac{1}{2}ab$, so four triangles give $4 \times \frac{1}{2}ab = 2ab$.
4. The area of the inner square is $c^2$.
5. Equating the two area expressions gives:

$$a^2 + 2ab + b^2 = 2ab + c^2$$

6. Subtracting $2ab$ from both sides results in:

$$a^2 + b^2 = c^2$$

This proves the relationship between the sides of a right-angled triangle.

Applications of Pythagoras Theorem

The Pythagoras Theorem is a versatile tool used extensively in both abstract geometry and practical problem-solving:

• Calculating Unknown Lengths: Finding a missing side of a right-angled triangle when two sides are known.
• Checking for Right Angles: Using the converse of the theorem. If the side lengths of a triangle satisfy $a^2 + b^2 = c^2$, then the triangle is guaranteed to be right-angled.
• Real-World Distances: Solving spatial problems, such as determining the length of ladders leaning against walls, finding the shortest diagonal path across a rectangular field, or calculating direct distances in navigation.
• Foundation for Other Math Concepts: It serves as the basis for the distance formula in coordinate geometry and the fundamental trigonometric identity $\sin^2\theta + \cos^2\theta = 1$.

Worked Examples

Example 1: Finding the Hypotenuse In a right-angled triangle $PQR$, the angle at $Q$ is $90^\circ$. If $PQ = 6\text{ cm}$ and $QR = 8\text{ cm}$, find the length of $PR$.

Solution: Since the angle at $Q$ is $90^\circ$, $PR$ is the hypotenuse. Let $PR = c$, $PQ = a = 6\text{ cm}$, and $QR = b = 8\text{ cm}$. By Pythagoras theorem: $$c^2 = a^2 + b^2$$ $$c^2 = 6^2 + 8^2$$ $$c^2 = 36 + 64$$ $$c^2 = 100$$ Taking the square root of both sides: $$c = \sqrt{100} = 10\text{ cm}$$ The length of $PR$ is $10\text{ cm}$.

Example 2: Finding a Shorter Side A ladder $13\text{ m}$ long leans against a vertical wall. The foot of the ladder is $5\text{ m}$ away from the base of the wall. How high up the wall does the ladder reach?

Solution: The ladder acts as the hypotenuse ($c = 13\text{ m}$). The distance from the base is one leg ($a = 5\text{ m}$), and the height on the wall is the other leg ($b$). Using Pythagoras theorem: $$a^2 + b^2 = c^2$$ $$5^2 + b^2 = 13^2$$ $$25 + b^2 = 169$$ $$b^2 = 169 - 25$$ $$b^2 = 144$$ $$b = \sqrt{144} = 12\text{ m}$$ The ladder reaches a height of $12\text{ m}$ on the wall.

NECTA Exam Focus

While standalone direct questions strictly asking for the Pythagoras theorem can occasionally appear, NECTA CSEE predominantly tests this topic as an embedded requirement within broader mathematical problems.

Recurring Themes:

• Trigonometry and Vectors: Students frequently need to use Pythagoras theorem to find a missing side of a triangle before they can calculate sine, cosine, tangent, or the magnitude of a vector.
• Three-Dimensional Geometry: A common challenging problem type involves finding the diagonal of a cuboid (e.g., the length of a line connecting opposite corners of a room), which requires applying the Pythagoras theorem twice.
• Coordinate Geometry: Deriving or applying the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ fundamentally relies on this theorem.
• Word Problems: Scenario-based questions involving navigation (traveling North then East), shadows, guy-wires attached to poles, and rectangles with diagonal paths.

Common Pitfalls:

• Misidentifying the Hypotenuse: Students sometimes mistakenly assign the longest value to a leg instead of the hypotenuse, especially when the triangle is rotated.
• Incomplete Calculation: Forgetting to take the square root at the final step (e.g., stopping at $x^2 = 25$ instead of finishing with $x = 5$).
• Incorrect Arithmetic Operations: Adding the squares when looking for a shorter leg (instead of subtracting), or subtracting when looking for the hypotenuse.

Practice Problems

1. A rectangular piece of land is $40\text{ m}$ long and $30\text{ m}$ wide. A straight path is made from one corner of the land to the opposite corner. What is the length of this path?
2. In triangle $XYZ$, $\angle Y = 90^\circ$. If $XZ = 17\text{ cm}$ and $XY = 15\text{ cm}$, find the length of $YZ$.
3. A ship sails $24\text{ km}$ due South and then $10\text{ km}$ due West. How far is the ship from its starting point?
4. A tent is supported by a rope tied from its top to the ground. If the tent is $8\text{ m}$ tall and the rope is tied $6\text{ m}$ away from the base of the tent, what is the length of the rope?

Subtopics

• Proof of Pythagoras theorem
• Applications of Pythagoras theorem

Crosswalk Notes

Cross-version relationships are drafted in data/curricula/crosswalks/csee-basic-mathematics-2005-to-mathematics-2023.json. Partial and 2005-only mappings remain reviewable.