Pythagoras Theorem & its Converse

The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the sides of a right-angled triangle. It’s a cornerstone for understanding distances, areas, and trigonometry. The theorem provides a way to calculate the length of a side of a right triangle if you know the lengths of the other two sides. Its converse allows us to determine if a triangle is right-angled, given the lengths of its sides.

Formulae

Pythagorean Theorem: In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs or cathetus).

This can be expressed mathematically as: $a^2 + b^2 = c^2$ where:

• $a$ and $b$ are the lengths of the legs of the right triangle.
• $c$ is the length of the hypotenuse.

Converse of the Pythagorean Theorem: 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, then the triangle is a right-angled triangle.

Examples

Example-1: A right triangle has legs of length 3 cm and 4 cm. Find the length of the hypotenuse.

Using the Pythagorean Theorem: $a^2 + b^2 = c^2$ $3^2 + 4^2 = c^2$ $9 + 16 = c^2$ $25 = c^2$ $c = \sqrt{25} = 5$ cm. The hypotenuse is 5 cm long.

Example-2: A triangle has sides of length 5, 12, and 13. Determine if it’s a right triangle.

Using the Converse of the Pythagorean Theorem: $5^2 + 12^2 = 25 + 144 = 169$ $13^2 = 169$ Since $5^2 + 12^2 = 13^2$, the triangle is a right triangle.

Theorem with Proof

Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs).

Proof: There are several ways to prove the Pythagorean Theorem. Here’s a common visual proof using areas:

1. Start with a square with side length $(a + b)$.
2. Inside this square, draw four congruent right triangles, each with legs of length $a$ and $b$, and hypotenuse $c$. Arrange these triangles so they form another square inside, with side length $c$.
3. The area of the large square is $(a + b)^2$.
4. The area of the large square can also be calculated by summing the areas of the four triangles and the inner square: $4 * (1/2)ab + c^2$.
5. Therefore, $(a + b)^2 = 2ab + c^2$.
6. Expanding $(a + b)^2$, we get $a^2 + 2ab + b^2 = 2ab + c^2$.
7. Subtracting $2ab$ from both sides, we get $a^2 + b^2 = c^2$.

Common mistakes by students

Students often make the following mistakes:

• Incorrectly identifying the hypotenuse (always the side opposite the right angle).
• Forgetting to square the side lengths before adding them.
• Incorrectly applying the theorem to non-right triangles. The Pythagorean theorem applies *only* to right triangles.
• Confusing the Converse with the Theorem itself (e.g., assuming a triangle is right-angled without verifying the side lengths).
• Incorrectly solving for a side length after applying the formula, for example forgetting to take the square root at the end.

Real Life Application

The Pythagorean Theorem has many real-world applications:

• Construction: Ensuring right angles for building structures (e.g., using the 3-4-5 rule to check if a corner is a right angle).
• Navigation: Calculating distances (e.g., the shortest distance to a destination).
• Engineering: Designing bridges, buildings, and other structures.
• Surveying: Measuring distances and areas.
• Computer Graphics: Used in 2D and 3D graphics to calculate distances and positions.

Fun Fact

The Pythagorean Theorem was known long before Pythagoras. Evidence suggests that the Babylonians and Egyptians used the theorem and its principles centuries before Pythagoras’ time.

Recommended YouTube Videos for Deeper Understanding

Q.1 In a right-angled triangle, the hypotenuse is 13 cm and one side is 12 cm. What is the length of the other side?

Check Solution

Ans: A

Using the Pythagorean theorem, $a^2 + b^2 = c^2$. We know $c = 13$ and $a = 12$. Therefore, $12^2 + b^2 = 13^2$, or $144 + b^2 = 169$. Solving for $b$ gives $b = \sqrt{169 – 144} = \sqrt{25} = 5$.

Q.2 The sides of a triangle are 8 cm, 15 cm, and 17 cm. Is this a right-angled triangle?

Check Solution

Ans: A

Check if $a^2 + b^2 = c^2$. $8^2 + 15^2 = 64 + 225 = 289$. Also, $17^2 = 289$. Since $8^2 + 15^2 = 17^2$, the triangle is a right-angled triangle.

Q.3 A ladder of 20 feet leans against a wall. The foot of the ladder is 12 feet away from the wall. How high up the wall does the ladder reach?

Check Solution

Ans: B

Using the Pythagorean theorem, $a^2 + b^2 = c^2$. The ladder is the hypotenuse ($c = 20$), and the distance from the wall is one side ($a = 12$). Therefore, $12^2 + b^2 = 20^2$, or $144 + b^2 = 400$. Solving for $b$ gives $b = \sqrt{400 – 144} = \sqrt{256} = 16$.

Q.4 Which of the following sets of numbers cannot represent the sides of a right-angled triangle?

Check Solution

Ans: D

Check if $a^2 + b^2 = c^2$ for each set. A) $3^2 + 4^2 = 9 + 16 = 25 = 5^2$. B) $5^2 + 12^2 = 25 + 144 = 169 = 13^2$. C) $7^2 + 24^2 = 49 + 576 = 625 = 25^2$. D) $2^2 + 3^2 = 4 + 9 = 13$. Since $6^2 = 36$ and $13 \ne 36$, this is not a right-angled triangle.

Q.5 In a right-angled triangle, if the two shorter sides are $x$ and $x+7$, and the hypotenuse is $x+8$, what is the value of $x$?

Check Solution

Ans: A

Using the Pythagorean theorem, $x^2 + (x+7)^2 = (x+8)^2$. Expanding, $x^2 + x^2 + 14x + 49 = x^2 + 16x + 64$. Simplifying, $x^2 – 2x – 15 = 0$. Factoring, $(x-5)(x+3) = 0$. Since side lengths cannot be negative, $x = 5$.

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