Problem 32 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The lengths 7, 24 and 26 are sides of a right triangle because the square of the longest side is equal to the sum of squares of the other two sides (\(7^2 + 24^2 = 26^2\)).

Step 1: Identification

From the given lengths, identify the longest side. Here the longest length is 26.

Step 2: Apply Pythagorean theorem

Apply the Pythagorean theorem by squaring the lengths of the two shorter sides and adding them together. Also square the longest side. Calculate \(7^2 + 24^2\) and \(26^2\).

Step 3: Compare the values

Compare the values obtained. If the sum of squares of the two shorter sides is equal to the square of the longest side, those lengths form a right triangle. If not, they don't form a right triangle.

Key Concepts

Right Triangle

A right triangle is one of the simplest forms of triangles in geometry. It features one angle that is exactly 90 degrees. In any triangle, the side opposite to the right angle is the longest side and is called the hypotenuse. The other two sides are called the legs. The special property that defines right triangles is the Pythagorean Theorem. This theorem states that for a triangle to be classified as a right triangle, the sum of the squares of the lengths of the two shorter sides (legs) should equal the square of the length of the longest side (hypotenuse). For example, with sides 7, 24, and 26, you identify 26 as the hypotenuse and then apply the theorem:

• Calculate: \(7^2 + 24^2\) = 49 + 576 = 625
• Calculate: \(26^2\) = 676
• Compare the results to see if they match.

If they do equal, then the three sides you have do indeed create a right triangle. This understanding can easily help in various fields like construction and design where identifying right angles is essential.

Mathematical Reasoning

Mathematical reasoning is a critical part of solving problems across all branches of mathematics. It involves applying logic to figure out the properties and relationships between numbers and shapes. When determining whether specific side lengths can form a right triangle, we employ deductive reasoning.

• First, identify the potential hypotenuse, as in our example, 26.
• Check this by using the Pythagorean Theorem, squaring the sides and comparing the sums.
• Arrive at a conclusion based on whether both sides of the equation hold true.

By using these reasoning steps, we ensure each step is based on sound arithmetic and principles. This logical process helps not only in solving geometric problems but also nurtures strong analytical skills that are transferable to other areas of learning and problem-solving.

Triangular Inequality

The triangular inequality states that for any triangle, the sum of the lengths of any two sides must always be greater than or equal to the length of the remaining side. This property serves as a check when verifying potential triangle side lengths. For determining a right triangle:

• First, confirm that \(7 + 24 > 26\), \(7 + 26 > 24\), and \(24 + 26 > 7\).
• This ensures that the sides not only could form a triangle but also meet the initial conditions needed for all triangles.

If the inequalities hold true, one can proceed with checking right angle properties using the Pythagorean Theorem. Understanding triangular inequality helps guard against errors in considering non-feasible triangle formations. This foundational concept ensures clarity and correctness in geometry beyond just identifying right triangles.