Solving for Line Segment BC in a Right Angled Triangle

In the context of right-angled triangles, the Pythagorean theorem is a fundamental principle that allows us to find the length of an unknown side given the lengths of the other two. The 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 other two sides.

Problem Statement

A right-angled triangle ABC has the following properties:

Angle B is 90° The hypotenuse AC (side c) is 7.4 cm Another side AB (side a) is 6.3 cm

Our task is to find the length of the remaining line segment BC (side b).

Solution

The Pythagorean theorem can be mathematically represented as:

( c^2 a^2 b^2 )

Where:

( c ) is the hypotenuse ( a ) is one of the other two sides ( b ) is the remaining side

Step-by-Step Calculation

We start with the given values: ( c 7.4 ) cm (hypotenuse AC) ( a 6.3 ) cm (side AB) We need to find ( b ) (side BC).

Using the Pythagorean theorem:

( 7.4^2 6.3^2 b^2 )

Calculating the squares:

( 54.76 39.69 b^2 )

Subtract 39.69 from both sides:

( b^2 54.76 - 39.69 )

( b^2 15.07 )

Take the square root of both sides:

( b sqrt{15.07} )

( b ≈ 3.88 ) cm (side BC)

Implications and Further Learning

This method is not only useful in geometric problems but also in various real-world applications, such as construction, navigation, and engineering. Understanding the Pythagorean theorem is crucial for solving a wide range of problems related to distances and angles.

Conclusion

The length of the line segment BC in the right-angled triangle ABC, given that angle B is 90°, AC is 7.4 cm, and AB is 6.3 cm, is approximately 3.88 cm.