Applying the Pythagorean Theorem to the Problem

Oct 13, 2025 by miodeya.com 63 views

Introduction

In geometry, a right triangle is a triangle with one angle that is 90 degrees. The hypotenuse of a right triangle is the side opposite the right angle, and it is always the longest side. In this problem, we are given a right triangle with a hypotenuse of 13 feet, and one of the legs needs to be 7 feet longer than the other leg. We need to find the length of the shorter leg, denoted by $x$ . In this article, we will use the Pythagorean theorem to solve this problem.

The Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that relates the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be expressed as:

a^2 + b^2 = c^2

where $a$ and $b$ are the lengths of the two legs, and $c$ is the length of the hypotenuse.

Applying the Pythagorean Theorem to the Problem

In this problem, we are given that the hypotenuse is 13 feet, and one of the legs needs to be 7 feet longer than the other leg. Let's denote the length of the shorter leg as $x$ . Then, the length of the longer leg is $x + 7$ . We can now apply the Pythagorean theorem to this problem:

x^2 + (x + 7)^2 = 13^2

Simplifying the Equation

To simplify the equation, we can expand the squared term:

x^2 + x^2 + 14x + 49 = 169

Combine like terms:

2x^2 + 14x + 49 = 169

Subtract 169 from both sides:

2x^2 + 14x - 120 = 0

Solving the Quadratic Equation

The equation we obtained is a quadratic equation in the form $ax^2 + bx + c = 0$ . We can solve this equation using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 2$ , $b = 14$ , and $c = -120$ . Plugging these values into the formula, we get:

x = \frac{-14 \pm \sqrt{14^2 - 4(2)(-120)}}{2(2)}

Simplify the expression under the square root:

x = \frac{-14 \pm \sqrt{196 + 960}}{4}

x = \frac{-14 \pm \sqrt{1156}}{4}

x = \frac{-14 \pm 34}{4}

Finding the Length of the Shorter Leg

We obtained two possible values for $x$ :

x = \frac{-14 + 34}{4} = \frac{20}{4} = 5

x = \frac{-14 - 34}{4} = \frac{-48}{4} = -12

Since the length of a leg cannot be negative, we discard the negative solution. Therefore, the length of the shorter leg is $x = 5$ feet.

Conclusion

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that relates the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I use the Pythagorean theorem to solve a problem?

A: To use the Pythagorean theorem, you need to know the lengths of the two legs and the hypotenuse of the right triangle. You can then plug these values into the equation $a^2 + b^2 = c^2$ , where $a$ and $b$ are the lengths of the two legs, and $c$ is the length of the hypotenuse.

Q: What if I only know the length of the hypotenuse and one of the legs?

A: If you only know the length of the hypotenuse and one of the legs, you can use the Pythagorean theorem to find the length of the other leg. Let's say you know the length of the hypotenuse is $c$ and the length of one of the legs is $a$ . You can then plug these values into the equation $a^2 + b^2 = c^2$ , where $b$ is the length of the other leg.

Q: How do I solve a quadratic equation that results from using the Pythagorean theorem?

A: If the Pythagorean theorem results in a quadratic equation, you can solve it using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . This formula will give you two possible solutions, but you need to check which one is valid in the context of the problem.

Q: What if I have a right triangle with a hypotenuse of 10 feet and one leg of 6 feet? How do I find the length of the other leg?

A: To find the length of the other leg, you can use the Pythagorean theorem: $a^2 + b^2 = c^2$ . Plugging in the values you know, you get $6^2 + b^2 = 10^2$ . Simplifying the equation, you get $36 + b^2 = 100$ . Subtracting 36 from both sides, you get $b^2 = 64$ . Taking the square root of both sides, you get $b = 8$ . Therefore, the length of the other leg is 8 feet.

Q: Can I use the Pythagorean theorem to find the length of the hypotenuse?

A: Yes, you can use the Pythagorean theorem to find the length of the hypotenuse. If you know the lengths of the two legs, you can plug these values into the equation $a^2 + b^2 = c^2$ , where $c$ is the length of the hypotenuse. Solving for $c$ , you can find the length of the hypotenuse.

Q: What are some real-world applications of the Pythagorean theorem?

A: The Pythagorean theorem has many real-world applications, including:

Building design: Architects use the Pythagorean theorem to design buildings and ensure that the walls and floors are perpendicular.
Engineering: Engineers use the Pythagorean theorem to calculate the stress and strain on materials in structures such as bridges and buildings.
Navigation: Pilots and sailors use the Pythagorean theorem to calculate distances and directions.
Physics: Physicists use the Pythagorean theorem to calculate the trajectory of projectiles and the motion of objects.

Conclusion

The Pythagorean theorem is a powerful tool for solving problems involving right triangles. By understanding how to use the theorem and how to solve quadratic equations, you can apply it to a wide range of real-world problems. Whether you're an architect, engineer, or physicist, the Pythagorean theorem is an essential tool for your work.