Problem 73 - 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 triangle with sides 2, 4, and 5 is not a right-angled triangle.

Step 1: Understand the Pythagorean Theorem

The Pythagorean theorem states that in 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 \( b \). The equation is: \(a^2 + b^2 = c^2\).

Step 2: Substitute the Given Values

Substitute \( a = 2 \), \( b = 4 \), and \( c = 5 \) into the Pythagorean theorem equation: \(2^2 + 4^2 = 5^2\).

Step 3: Simplify the Left Side

Calculate \(2^2 + 4^2\). This equals \(4 + 16\), which simplifies to \(20\).

Step 4: Compare with the Right Side

Calculate \(5^2\), which equals \(25\). Now compare both sides: \(20\) and \(25\).

Step 5: Analyze the Result

Since \(20 eq 25\), the statement is false. This implies that the triangle with sides \(2, 4, 5\) cannot be a right-angled triangle according to the Pythagorean theorem.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring 90 degrees. This angle is called a right angle. The side opposite this right angle is known as the hypotenuse, which is the longest side of the triangle. The other two sides are referred to as the legs of the triangle.
Understanding right triangles is essential because they are the foundation of the Pythagorean theorem. This theorem provides a relationship between the sides of a right triangle, allowing us to calculate distances and solve many problems involving right-angled situations.
In practical terms, right triangles are present in architecture, navigation, and many forms of engineering. They serve as crucial elements in the design and analysis of various structures.

Hypotenuse

The hypotenuse is the side of a right triangle that is always opposite the right angle. It is the longest side since it spans across the widest part of the triangle.
When you apply the Pythagorean theorem, it is the hypotenuse that you solve for, using the formula: \[ a^2 + b^2 = c^2 \] where \( c \) represents the hypotenuse, and \( a \) and \( b \) are the other two sides.
Knowing how to identify and calculate the hypotenuse is crucial in geometry, where it assists in finding lengths in right-angled structures. Calculating the hypotenuse is fundamental in applications such as computing displacement vectors or designing ramps and stairs.

Triangle Sides

The sides of a triangle have specific roles in determining its type and properties. In right triangles, one key aspect is the relationship formed by the Pythagorean theorem.
Looking closer at the exercise provided, we see three potential side lengths: 2, 4, and 5. When you substitute these into the formula:\[ 2^2 + 4^2 = 5^2 \] and simplify, the left-hand expression yields 20, while the right-hand side yields 25. This imbalance suggests that these side lengths do not fit the criteria of a right triangle.
This exercise helps underscore the importance of checking that a potential set of side lengths does indeed satisfy the Pythagorean theorem to qualify as a right triangle.

Mathematical Proof

Mathematical proof is a logical argument that verifies the truth of a mathematical proposition. It often follows a structured process where established principles are applied to derive conclusions.
In the case of the Pythagorean theorem, mathematical proof involves substituting known values into the formula to verify if all conditions are met. When the sides 2, 4, and 5 are tested:\[ 2^2 + 4^2 = 5^2 \]The calculation results in a false statement with 20 not equaling 25. This proof or disproof process helps in identifying whether given data conform to specific theorems.
Proof not only supports understanding in geometry but also plays a vital role in validating complex mathematical theories and real-world problem-solving.