Problem 85 - Elementary and Intermediate Algebra

Question

When the lengths of the sides of a certain triangle are substituted into the equation of the Pythagorean theorem, the result is a false statement. Explain why. $$ \begin{aligned} a^{2}+b^{2} &=c^{2} \\ 2^{2}+4^{2} &=5^{2} \\ 4+16 &=25 \\\ 20 &=25 \quad \text { False } \end{aligned} $$

Step-by-Step Solution

Answer

The sides 2, 4, and 5 do not satisfy the Pythagorean theorem, so the triangle is not right-angled.

Step 1: Understand the Pythagorean Theorem

The Pythagorean Theorem states that for a right-angled triangle, the square of the length of the hypotenuse \(c\) is equal to the sum of the squares of the lengths of the other two sides \(a\) and \(b\). This can be expressed as \(a^2 + b^2 = c^2\).

Step 2: Substitute the Given Values

Substitute the given side lengths into the equation \(a^2 + b^2 = c^2\). For this problem, \(a = 2\), \(b = 4\), and \(c = 5\). Thus, substitute as follows: \(2^2 + 4^2 = 5^2\).

Step 3: Evaluate the Left Side of the Equation

Compute \(2^2 + 4^2\). First, \(2^2 = 4\) and \(4^2 = 16\). So, \(4 + 16 = 20\).

Step 4: Evaluate the Right Side of the Equation

Compute \(5^2\), which is \(25\).

Step 5: Compare Both Sides of the Equation

Compare the results of the left side \((20)\) and the right side \((25)\). 20 is not equal to 25.

Step 6: Conclusion

Since the equation \(20 = 25\) is a false statement, the triangle with sides 2, 4, and 5 is not a right-angled triangle, as they don't satisfy the Pythagorean theorem.

Key Concepts

Right-Angled Triangle

A right-angled triangle is a special type of triangle where one of its angles measures exactly 90 degrees. This right angle implies a unique relationship between the lengths of its sides. The side opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides are referred to as the legs.

The Pythagorean Theorem specifically applies to right-angled triangles. It's a mathematical principle that tells us if we square the lengths of the two shorter sides and add them together, this sum will equal the square of the hypotenuse. For example, if we have a triangle with legs of lengths 'a' and 'b', and hypotenuse 'c', the theorem is expressed as:

• \( a^2 + b^2 = c^2 \)

Use Pythagorean Theorem as a test: if your triangle does not satisfy this equation, it simply cannot include a 90-degree angle.

False Statement

A false statement is an assertion that is not correct or true. In mathematics, when evaluating expressions, any inequality naturally arises as a false statement. In the problem, the lengths 2, 4, and 5 were put into the Pythagorean Theorem equation. However, the result was

• From the left side: \( 2^2 + 4^2 = 20 \)
• From the right side: \( 5^2 = 25 \)

Both sides of the equation turned out unequal, forming the false statement \( 20 eq 25 \). Consequently, the lengths 2, 4, and 5 cannot describe a right-angled triangle, as they don't uphold the equation of the Pythagorean Theorem.

Mathematical Proof

Mathematical proof is a demonstration of the truth of a proposition or statement, using logical reasoning and accepted truths or axioms. A proof establishes the validity of a theorem or statement beyond any doubt. For the Pythagorean Theorem, its proof guarantees that if the equation holds true, the triangle is right-angled.Let’s break down the process: by inserting specific values into the equation, and finding both sides of the equation not equal, we effectively completed a proof that shows these values do not work for a right-angled triangle. This specific exercise of substituting numbers like 2, 4, and 5 into the theorem's formula is part of a modest style of proof, often called a contradiction. Here, the incorrect equality

• \( 20 = 25 \)

was reached, clearly showcasing the proposition was false. In this context, proving a false statement generated a valid conclusion - confirming these side lengths do not form a right-angled triangle.