Problem 28 - Algebra 1

Question

Determine whether the given lengths are sides of a right triangle. Explain your reasoning. $$2,10,11$$

Step-by-Step Solution

Answer

No, 2, 10, and 11 cannot be lengths of the sides of a right triangle based on the Pythagorean theorem.

Step 1: Identify the longest side

From the given lengths 2, 10, and 11, it can be seen that 11 is the longest. Therefore, if this is a right triangle, then 11 should be the hypotenuse.

Step 2: Squaring the sides and applying Pythagorean theorem

Next, calculate the square of each side length. The squares of the sides are \(2^2 = 4\), \(10^2 = 100\), and \(11^2 = 121\). The Pythagorean theorem states that \(a^2 + b^2 = c^2\), where c represents the hypotenuse and a, b are the lengths of the other two sides. Substituting the actual values, we get the equation \(4 + 100 = 121\). However, this simplifies as 104 which does not equal 121.

Step 3: Conclusion

Since \(a^2 + b^2\) does not equal \(c^2\), we can conclude that the given lengths cannot form a right triangle according to the Pythagorean theorem.

Key Concepts

Right Triangle

A right triangle is one of the most fundamental shapes in geometry, characterized by one angle measuring exactly 90 degrees. This type of triangle has a wide application, from architecture to trigonometry, and is the basis for the Pythagorean theorem. A right triangle will always have two sides that meet at the right angle, known as 'legs', and the longest side opposite the right angle, called the 'hypotenuse'.

Understanding the properties of right triangles is essential for solving problems related to geometry and trigonometry. For instance, in the problem where lengths of 2, 10, and 11 are given, one can apply knowledge of right triangles to determine whether these lengths can constitute the sides of a right triangle.

Hypotenuse

The hypotenuse is the longest side of a right triangle, directly opposite the right angle. It plays a critical role in the famous Pythagorean theorem and can be distinctly identified in any given right triangle. In the context of our exercise, among the side lengths provided, 11 is identified as the potential hypotenuse because it is the longest.

The length of the hypotenuse is particularly useful, not only because it connects with the theorem but also it is the basis for defining the sine, cosine, and tangent functions in trigonometry. Calculating the length of the hypotenuse or checking the correctness of given triangle sides, like in our exercise, requires consistent application of the Pythagorean theorem.

Geometric Proof

A geometric proof is a logical argument to demonstrate the truth of a geometric statement, using a structured series of statements. Each statement in the proof builds upon the previous ones. A classical example is proving whether a set of side lengths can form a right triangle using the Pythagorean theorem. The process usually involves identifying the given lengths, applying the theorem, which is represented mathematically as \(a^2 + b^2 = c^2\), and drawing a logical conclusion based on the calculation.

In our exercise, the geometric proof involves squaring the given side lengths, summing the squares of the shorter sides, and then checking whether this sum equals the square of the longest side (potential hypotenuse). The conclusion drawn is that the given lengths do not satisfy the Pythagorean theorem, and thus, they cannot form a right triangle.