Finding the Base of a Right-Angled Triangle Using the Pythagorean Theorem

When working with right-angled triangles, the Pythagorean Theorem is a powerful tool for solving for unknown sides. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (legs). The formula is expressed as:

[c^2 a^2 b^2]

Where (c) is the hypotenuse, (a) is one leg, and (b) is the other leg.

Problem Statement

Given a right-angled triangle where the hypotenuse is 26 cm and one leg (perpendicular) is 24 cm, we need to find the length of the other leg (base).

Using the Pythagorean Theorem

Starting with the formula:

[26^2 24^2 b^2]

First, calculate the squares:

[676 576 b^2]

Next, isolate (b^2) by subtracting 576 from both sides:

[100 b^2]

Finally, take the square root of both sides:

[b sqrt{100} 10 text{ cm}]

This means the length of the base is 10 cm.

Another Approach Using the Pythagorean Theorem

We can also use algebraic steps to solve for the base:

[26^2 24^2 x^2]

Calculate the squares:

[676 576 x^2]

Isolate (x^2) by subtracting 576 from both sides:

[x^2 100]

Take the square root of both sides:

[x sqrt{100} 10 text{ cm}]

A negative length would be meaningless in the context of a physical measurement, so we discard (-10) as a solution.

Mathematical Insight

This problem also shows a relationship between 3:4:5 triangles. Here, the given sides 24 and 26 relate to a triangle with sides in the ratio 3:4:5. A quick check confirms:

[24 3 times 8]

[26 5 times 5.2 text{ (approximately 10 times 2.6 to get 26)}]

Therefore, the base in this triangle should be 10 cm, which aligns with our calculated result.

Conclusion

By using the Pythagorean Theorem, we can easily find the length of any side in a right-angled triangle. In this case, we determined that the base of the triangle is 10 cm. Understanding and applying the theorem is crucial for any student of geometry and can be of significant practical use in fields such as engineering, architecture, and physics.