To find the base of a right-angled triangle, you primarily use the Pythagorean theorem, which relates the lengths of the three sides. This fundamental theorem allows you to calculate an unknown side if the lengths of the other two sides are known.
Understanding the Pythagorean Theorem
The Pythagorean theorem states that 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 (the base and the height).
Mathematically, this is expressed as:
a² + b² = c²
Where:
arepresents the height (or one of the legs) of the triangle.brepresents the base (or the other leg) of the triangle.crepresents the hypotenuse (the longest side).
Key Components of a Right-Angled Triangle
| Component | Description |
|---|---|
| Base | One of the two sides that form the right angle, typically the bottom side. |
| Height | The other side that forms the right angle, perpendicular to the base. |
| Hypotenuse | The side opposite the right angle; it is always the longest side. |
Formula to Find the Base
If you know the length of the hypotenuse (c) and the height (a) of a right-angled triangle, you can find the length of its base (b) by rearranging the Pythagorean theorem.
From a² + b² = c², we can isolate b²:
b² = c² - a²
To find b (the base), you then take the square root of both sides:
b = √(c² - a²)
This formula allows you to directly calculate the base when the other two sides are known.
Steps to Calculate the Base
Follow these simple steps to find the base of a right-angled triangle:
- Identify the knowns: Determine the lengths of the hypotenuse (
c) and the height (a). - Square the hypotenuse: Calculate
c². - Square the height: Calculate
a². - Subtract the squared height from the squared hypotenuse: Perform
c² - a². This result gives youb². - Take the square root: Find the square root of the result from step 4 to get the length of the base (
b).
Example Calculation
Let's say you have a right-angled triangle with:
- Hypotenuse (c) = 10 units
- Height (a) = 6 units
Now, let's find the base (b):
- Identify knowns:
c = 10,a = 6. - Square the hypotenuse:
c² = 10² = 100. - Square the height:
a² = 6² = 36. - Subtract:
b² = c² - a² = 100 - 36 = 64. - Take the square root:
b = √64 = 8.
Therefore, the base of the right-angled triangle is 8 units.
By applying the Pythagorean theorem and its derived formula, you can accurately determine the base of any right-angled triangle given the lengths of its hypotenuse and height.
