Introduction
- The Pythagorean Theorem is a simple rule that helps us figure out the length of one side in a right-angled triangle if we know the other two.
- It is one of the most fundamental and well-known principles in geometry.
- It is widely used in mathematics, physics, engineering, and everyday problem-solving to calculate distances or unknown side lengths.
Real-Life Application:
What is Pythagoras Theorem?
- In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Mathematically, if a right-angled triangle has sides of lengths $a$, $b$, and $c$ (where $c$ is the hypotenuse), then:
$$c^2 = a^2 + b^2$$
Key points:
- Applies Only to Right-Angled Triangles
- Hypotenuse is the Longest Side
- It helps to find the length of any side when the other two are known.
How To Find The Length of The Missing Side in a Right-Angled Triangle?
- The Pythagorean Theorem helps find a missing side in a right-angled triangle when two sides are known.
Steps to Find Missing Sides in a Right-Angled Triangle:
1
Identify the given sides
2
Use the formula
3
Plug the values
4
Solve for the unknown side.
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Solved Example
A triangle has one side 6 cm long, another side 8 cm long, and a right angle between them. What is the length of the hypotenuse?
SOLUTION
1
Given:
- Base $a = 6$ cm
- Height $b = 8$ cm
2
Using the formula:
$$c^2 = a^2 + b^2$$
3
Plug the values and solve:
$$c^2 = 6^2 + 8^2$$
$$c^2 = 36 + 64$$
$$c^2 = 100$$
$$c = \sqrt{100} = 10$$
Final Answer: $c = 10$ cm
Solved Example
A right-angled triangle has one side 9 cm and another side 12 cm. What is the length of the hypotenuse?
SOLUTION
1
Identify the Given Sides:
- Base $a = 9$ cm
- Height $b = 12$ cm
2
Use The Formula:
$$c^2 = a^2 + b^2$$
3
Plug the values:
$$c^2 = 9^2 + 12^2$$
4
Solve for the unknown side:
$$c^2 = 81 + 144$$
$$c^2 = 225$$
$$c = \sqrt{225} = 15$$
Final Answer: $c = 15$ cm
Solved Example
In a right-angled triangle, the base is 7 cm and the height is 24 cm. Find the length of the hypotenuse.
SOLUTION
1
Identify the Given Sides:
- Base $a = 7$ cm
- Height $b = 24$ cm
2
Use The Formula:
$$c^2 = a^2 + b^2$$
3
Plug the values:
$$c^2 = 7^2 + 24^2$$
4
Solve for the unknown side:
$$c^2 = 49 + 576$$
$$c^2 = 625$$
$$c = \sqrt{625} = 25$$
Final Answer: $c = 25$ cm
Solved Example
A right-angled triangle has a hypotenuse of 13 cm and one side of 5 cm. What is the length of the other side?
SOLUTION
1
Identify the Given Sides:
- Base $a = 5$ cm
- Hypotenuse $c = 13$ cm
2
Use The Formula and rearrange it:
$$c^2 = a^2 + b^2$$
$$b^2 = c^2 – a^2$$
3
Plug the values:
$$b^2 = 13^2 – 5^2$$
4
Solve for the unknown side:
$$b^2 = 169 – 25$$
$$b^2 = 144$$
$$b = \sqrt{144} = 12$$
Final Answer: $b = 12$ cm
