Finding the Length of BC in a Right Angled Triangle

Understanding Right-Angled Triangles

A right-angled triangle, also known as a right triangle, is a triangle in which one angle is 90 degrees. In any right-angled triangle, the side opposite the right angle, known as the hypotenuse, is the longest side.

Given a right-angled triangle ABC, where angle C is the right angle, we can use the Pythagorean theorem to find the length of any side. The Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Applying the Pythagorean Theorem

In this problem, we are given that AB is the hypotenuse of the right-angled triangle ABC, with AB25 cm and AC7 cm. We need to determine the length of BC.

Let's solve for BC using the formula:

( AB^2 BC^2 AC^2 )

Substituting the given values:

( 25^2 BC^2 7^2 )

Calculating the squares:

( 625 BC^2 49 )

Subtracting 49 from both sides:

( BC^2 625 - 49 )

( BC^2 576 )

Taking the square root of both sides:

( BC sqrt{576} )

( BC 24 ) cm

Verification

To ensure the calculation is accurate, let's verify the value using different methods. For example, we can use the same formula with different labelings of sides:

( 26^2 10^2 BC^2 )

Calculating the squares:

( 676 100 BC^2 )

Subtracting 100 from both sides:

( BC^2 676 - 100 )

( BC^2 576 )

Again, taking the square root:

( BC sqrt{576} )

( BC 24 ) cm

Further simplification of the square root operation can be done as follows:

( BC^2 576 )

( BC 24 ) cm

Conclusion

Therefore, the length of BC in the right-angled triangle ABC is 24 cm. This method ensures that the relationship between the sides of the triangle is maintained according to the Pythagorean theorem.

Useful Pythagorean Theorem Solve for:

Right Triangle:

1. Given AB25 cm, AC7 cm:

2. Given AB26 cm, AC10 cm:

In both cases, the length of BC can be found using the theorem.