How to Use Pythagoras' Theorem
Pythagoras' Theorem is an essential concept in Maths, particularly in right-angled triangles. It allows you to find the length of a side in a right-angled triangle when you know the lengths of the other two sides. This guide will explain what Pythagoras' Theorem is, how to use it, and provide worked examples to help you understand how to apply it in different situations.
What is Pythagoras' Theorem?
Pythagoras' Theorem applies to right-angled triangles, where one of the angles is exactly $90^\circ$. The theorem states that:
$$a^2 + b^2 = c^2$$
Where:
- $a$ and $b$ are the lengths of the two legs (the sides that form the right angle).
- $c$ is the length of the hypotenuse (the side opposite the right angle, which is always the longest side of the triangle).
Key Points:
- You can only use Pythagoras' Theorem with right-angled triangles.
- The hypotenuse is always the longest side of the triangle and is always opposite the right angle.
The video below explains how to use Pythagoras' Theorem in more detail.
How to Use Pythagoras' Theorem
To use Pythagoras' Theorem to find a missing side in a right-angled triangle, follow these steps:
Step 1: Identify the sides
- Identify which sides of the triangle you know. You'll need the lengths of two sides to use the theorem:
- If you know the two legs ($a$ and $b$), you can find the hypotenuse ($c$).
- If you know one leg ($a$ or $b$) and the hypotenuse ($c$), you can find the missing leg.
Step 2: Substitute the known values into the formula
- Plug the known values into the Pythagoras' Theorem equation: $a^2 + b^2 = c^2$.
Step 3: Solve for the unknown side
- If you're finding the hypotenuse, rearrange the formula to solve for $c$: $c = \sqrt{a^2 + b^2}$
- If you're finding one of the legs, rearrange the formula to solve for $a$ or $b$:
$$a = \sqrt{c^2 - b^2} \quad \text{or} \quad b = \sqrt{c^2 - a^2}$$
Step 4: Calculate the result
- After rearranging the formula, calculate the value of the unknown side.
Worked Examples
Example 1: Finding the Hypotenuse
In a right-angled triangle, one leg is 6 cm and the other leg is 8 cm. Find the length of the hypotenuse.
Using Pythagoras' Theorem:
$$a^2 + b^2 = c^2$$
Substitute the values:
$$6^2 + 8^2 = c^2$$
$$36 + 64 = c^2$$
$$100 = c^2$$
Now, take the square root of both sides:
$$c = \sqrt{100} = 10$$
So, the length of the hypotenuse is 10 cm.
Example 2: Finding a Missing Leg
In a right-angled triangle, the hypotenuse is 13 cm and one leg is 5 cm. Find the length of the other leg.
Using Pythagoras' Theorem:
$$a^2 + b^2 = c^2$$
Rearrange the formula to solve for $a$:
$$a^2 = c^2 - b^2$$
Substitute the known values:
$$a^2 = 13^2 - 5^2$$
$$a^2 = 169 – 25$$
$$a^2 = 144$$
Now, take the square root of both sides:
$$a = \sqrt{144} = 12$$
So, the length of the missing leg is 12 cm.
Applications of Pythagoras' Theorem
Pythagoras' Theorem can be used in a variety of situations, including:
- Finding distances: In geometry problems, Pythagoras' Theorem can help you find the distance between two points in a right-angled triangle.
- Solving real-life problems: You can use the theorem to calculate the height of a building, the length of a ladder needed to reach a certain height, or the diagonal length of a rectangular object.
Example 3: Finding the Diagonal of a Rectangle
If a rectangular garden has a length of 9 m and a width of 12 m, what is the length of the diagonal (the distance between two opposite corners)?
Think of the rectangle as two right-angled triangles, where the length and width are the legs, and the diagonal is the hypotenuse. Using Pythagoras' Theorem:
$$a^2+b^2=c^2$$
Substitute the values:
$$9^2 + 12^2 = c^2$$
$$81 + 144 = c^2$$
$$225 = c^2$$
Take the square root of both sides:
$$c = \sqrt{225} = 15$$
So, the length of the diagonal is 15 m.
Key Points to Remember
- Right-angled triangles only: Pythagoras' Theorem can only be used with right-angled triangles.
- Formula: $$a^2 + b^2 = c^2$$
- Hypotenuse: The hypotenuse is always opposite the right angle and is the longest side.
- Rearranging the formula: When finding a leg, rearrange the formula to $a = \sqrt{c^2 - b^2}$ or $b = \sqrt{c^2 - a^2}$
By mastering Pythagoras' Theorem, you'll be able to solve many problems in both geometry and practical applications in your GCSE Maths exams.
