Solving Pythagorean Theorem Problems: A Comprehensive Guide

When faced with a problem involving right triangles, the Pythagorean theorem becomes an invaluable tool. This article will explore a specific problem involving a right triangle and demonstrate how to solve it step by step. We will also discuss different approaches to solving similar problems, including the use of quadratic equations and recognizing patterns in simple triangles.

Problem 1: Given a Hypotenuse of 20 cm and a Difference of 4 cm

Consider a right triangle where the hypotenuse is 20 cm, and the difference between the lengths of the other two sides is 4 cm. To solve for the lengths of these sides, we can use the Pythagorean Theorem.

Pythagorean Theorem: (a^2 b^2 c^2)

Here, (c 20) and (a - b 4).

Step 1: Express (a) in terms of (b)

We can express one side in terms of the other: let (a b 4).

Step 2: Substitute into the Pythagorean Theorem

Substituting (a b 4) into the equation, we get:

( (b 4)^2 b^2 20^2 ) ( b^2 8b 16 b^2 400 ) ( 2b^2 8b 16 400 ) ( 2b^2 8b - 384 0 )

Step 3: Solve the Quadratic Equation

The quadratic equation can be solved using the quadratic formula:

[ b frac{-B pm sqrt{B^2 - 4AC}}{2A} ]

Here, ( A 2 ), ( B 8 ), and ( C -384 ).

( b frac{-8 pm sqrt{8^2 - 4 cdot 2 cdot -384}}{2 cdot 2} )

( b frac{-8 pm sqrt{64 3072}}{4} )

( b frac{-8 pm sqrt{3136}}{4} )

( b frac{-8 pm 56}{4} )

( b 12 ) or ( b -16 )

Since length cannot be negative, ( b 12 ) cm.

Step 4: Calculate the Length of the Other Side

( a b 4 12 4 16 ) cm.

Therefore, the lengths of the other two sides are 12 cm and 16 cm.

Problem 2: Recognizing Simple Patterns in Right Triangles

Another approach to solving similar problems involves recognizing patterns in simple right triangles. Consider a right triangle where the hypotenuse is 15 cm, and the difference between the lengths of the other two sides is 3 cm.

Step 1: Express the Sides

Let's set one side as ( x ) and the other as ( x - 3 ).

Step 2: Apply the Pythagorean Theorem

Using the Pythagorean Theorem:

[ x^2 (x - 3)^2 15^2 ]

( x^2 x^2 - 6x 9 225 )

( 2x^2 - 6x 9 225 )

( 2x^2 - 6x - 216 0 )

Step 3: Solve the Quadratic Equation

Dividing the equation by 2:

[ x^2 - 3x - 108 0 ]

Factoring the equation:

[ (x - 9)(x 12) 0 ]

So, ( x 9 ) or ( x -12 ).

Since length cannot be negative, ( x 9 ) cm.

Step 4: Calculate the Length of the Other Side

( x - 3 9 - 3 6 ) cm.

Therefore, the lengths of the other two sides are 6 cm and 9 cm.

Conclusion

This article has demonstrated two effective methods for solving problems involving right triangles and the Pythagorean theorem. Whether through quadratic equations or recognizing simple patterns, the Pythagorean theorem remains a powerful tool for solving geometric problems. Understanding these methods can help you tackle similar problems with ease.

For further reading and practice, you may explore related questions on platforms like Quora or seek resources from educational institutions such as The Open University.