Question

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$a=9, c=41$$

Short answer

The length of the missing side, 'b', of the right triangle is 40 units.

Step-by-step solution

Recognize the Theorem

The Pythagorean theorem will be used to resolve this problem because two sides of a right triangle are provided and the third is asked to find. This theorem basically can be formulated as \(a^2 + b^2 = c^2\).

Substitute Known Values

We know a (9) and c (41) so it will be substituted into the Pythagorean Theorem to find the value of b. The unknown variable will be solved by doing \(41^2 - 9^2\).

Calculate Answer

The equation simplifies to 1681 - 81, which equals to 1600.

Square Root

Do the square root of 1600, because we are solving for length 'b', not 'b squared'. The square root of 1600 is 40.

Key concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles measures exactly 90 degrees, resembling an 'L' shape. It is this right angle that sets the foundation for many principles in geometry, including the famous Pythagorean theorem.

In a right triangle, the sides that form the right angle are referred to as the 'legs', and the side opposite the right angle is known as the 'hypotenuse', which is always the longest side. The relationship between the lengths of the sides is essential for solving various mathematical problems dealing with right triangles.

Properties of a Right Triangle

Right triangles have certain properties that are consistent across all examples:

• The hypotenuse is opposite the right angle and is the longest side.
• The legs meet to form a 90-degree angle.
• The area can be determined by taking half the product of the lengths of the legs.
• It adheres to the Pythagorean theorem, allowing for the calculation of any side's length if the other two are known.

Hypotenuse

The hypotenuse plays a pivotal role in the dynamics of a right triangle. This side is not just the longest; it is the key to unlocking many mathematical problems associated with right triangles. The length of the hypotenuse is crucial when applying the Pythagorean theorem, which is a fundamental equation used in various branches of mathematics, physics, engineering, and beyond.

Understanding the Hypotenuse

A deeper look at the hypotenuse reveals why it's so crucial:

• It is directly used in calculating the lengths of the triangle's legs.
• In the context of right-angled triangles, the hypotenuse can determine distances in a more efficient way compared to other methods.
• The measure of the hypotenuse is used in trigonometry to find the sine, cosine, and tangent of the angles.

The calculation of the hypotenuse is often the final step in problems involving right triangles, indicating a satisfying resolution to a geometric puzzle.

Mathematical Problem-Solving

Mathematical problem-solving is a critical skill that goes beyond pure math—it encourages logical reasoning and systematic thinking. When faced with a mathematical challenge, it's important to have a methodical approach.

Steps in Mathematical Problem-Solving

The typical steps to tackle a problem efficiently can be outlined as follows:

• Understanding the problem: Grasping the requirements and the end goal.
• Devising a plan: Determining which mathematical principles or theorems to apply, such as the Pythagorean theorem for right triangle problems.
• Carrying out the plan: Performing the actual calculations and manipulations.
• Reviewing the solution: Checking the results for accuracy and reasonableness.

In the case of finding a missing side of a right triangle, using the Pythagorean theorem epitomizes mathematical problem-solving. By isolating the unknown variable after recognizing the theorem's applicability and performing the necessary computations, the solution is obtained not by chance, but through a clear, logical progression of steps.