Solving for x in a Right Triangle: A Guide to Using the Pythagorean Theorem

When dealing with a right triangle, the Pythagorean theorem is a fundamental tool for solving various problems, including finding unknown sides or angles. This article will guide you through solving for the unknown variable x in a specific right triangle with given legs and hypotenuse.

Understanding the Problem

Consider a right triangle where the two legs have lengths of 15 feet and 36 feet, respectively. The length of the hypotenuse is given as 13x feet. Our goal is to find the value of x.

Using the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as:

[a^2 b^2 c^2]

where (a) and (b) are the lengths of the legs, and (c) is the length of the hypotenuse.

Applying the Theorem

Let’s apply this theorem to our specific triangle:

Substituting these values into the Pythagorean equation:

[15^2 36^2 (13x)^2]

Calculating the squares:

Summing these results:

[225 1296 1521]

Therefore, we have:

[1521 (13x)^2]

Solving for x

To solve for x, we take the square root of both sides:

[sqrt{1521} sqrt{(13x)^2}]

Calculating the square root:

[39 13x]

Dividing both sides by 13:

[x frac{39}{13} 3]

Thus, the value of x is

boxed{3}

Conclusion

The solution to the problem is x 3. The proof demonstrates how the Pythagorean theorem can be used to find unknown values in a right triangle. Understanding and applying the theorem is essential for solving various geometric problems and enhancing problem-solving skills in mathematics.