Problem 40 - Algebra 1: Concepts and Skills

Question

Determine whether the given lengths are sides of a right triangle. Explain your reasoning. $$ 7,24,26 $$

Step-by-Step Solution

Answer

No, the lengths 7, 24, and 26 do not form the sides of a right triangle.

Step 1: Identify the longest side (hypotenuse)

In this case, the longest side is 26. This will be considered as the hypotenuse.

Step 2: Apply the Pythagorean theorem

Calculate the square of the hypotenuse and the sum of the squares of the other two sides. The square of the hypotenuse is \(26^2 = 676\). The sum of the squares of the other two sides is \(7^2 + 24^2 = 49 + 576 = 625\).

Step 3: Compare the calculations

We need to check if the equation \(26^2 = 7^2 + 24^2\) holds true based on our calculations. While 26 squared equals 676, the sum of the squares of the other two sides equals 625. Since these two values are not equal, the lengths of 7, 24, and 26 do not form the sides of a right triangle.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry used to determine if a triangle with given side lengths is a right triangle. It relates the square of the length of the hypotenuse— the longest side of a right triangle — to the sum of the squares of the other two sides. This is expressed in the equation: \[ c^2 = a^2 + b^2 \] where \( c \) represents the hypotenuse, and \( a \) and \( b \) are the other two sides. To apply the theorem, follow these steps:

• Identify the three sides of the triangle.
• Determine which side is the longest to use as the hypotenuse (see next section: Hypotenuse Identification).
• Square the lengths of each side.
• Check if the sum of the squares of the two shorter sides equals the square of the hypotenuse.

If the equation holds true, the triangle is a right triangle; otherwise, it's not. This method is useful not only in proving mathematically whether a triangle is right-angled but also in ensuring accuracy in real-life applications.

Hypotenuse Identification

Identifying the hypotenuse correctly is crucial when using the Pythagorean Theorem. In any given triangle, the hypotenuse is always the longest side. In a right triangle, the hypotenuse is the side opposite the right angle. To identify the hypotenuse from a list of side lengths:

• Compare all three side lengths.
• Determine which has the greatest value.
• This side will be considered the hypotenuse.

For example, in the problem with sides 7, 24, and 26, the longest side is 26. Therefore, according to the Pythagorean Theorem, it is assumed to be the hypotenuse. This step is essential as using an incorrect hypotenuse in calculations will lead to inaccurate conclusions, rendering the application of the theorem invalid.

Triangle Sides Comparison

Triangle side comparison is important for determining if a set of three lengths can form a right triangle. Once the hypothesis side (the longest side) is identified, the next step is comparing the calculations. Let's look at the given side lengths: 7, 24, and 26.

• First, confirm that 26 is the hypotenuse as it is the longest side.
• Next, calculate the square of 26, which is 676.
• Then, calculate the sum of squares of the other two lengths: \(7^2 + 24^2 = 49 + 576 = 625\).
• Finally, compare 676 and 625.

Since they are not equal, it confirms that the side lengths do not fulfill the Pythagorean Theorem for a right triangle. By understanding and applying these comparisons effectively, one can determine the nature of the triangle with confidence, whether in academic contexts or practical situations.