Finding the Length of the Shorter Leg

Oct 13, 2025 by talynt.org 62 views

Introduction

In geometry, a right triangle is a triangle in which one angle is a right angle (90 degrees). In this article, we will explore a problem involving a right triangle with a specific condition. We are given that the shorter leg of the triangle is 4 feet shorter than the longer leg, and the longest side (hypotenuse) is $\sqrt{26}$ feet. Our goal is to find the lengths of the three sides of the triangle.

Understanding the Problem

Let's denote the length of the shorter leg as $x$ and the length of the longer leg as $y$ . Since the shorter leg is 4 feet shorter than the longer leg, we can write an equation:

x = y - 4

We are also given that the longest side (hypotenuse) is $\sqrt{26}$ feet. Using the Pythagorean theorem, we can write another equation:

x^2 + y^2 = (\sqrt{26})^2

Solving the Equations

We can substitute the expression for $x$ from the first equation into the second equation:

(y - 4)^2 + y^2 = 26

Expanding the left-hand side, we get:

y^2 - 8y + 16 + y^2 = 26

Combine like terms:

2y^2 - 8y - 10 = 0

Divide both sides by 2:

y^2 - 4y - 5 = 0

Now we have a quadratic equation in $y$ . We can solve it using the quadratic formula:

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

In this case, $a = 1$ , $b = -4$ , and $c = -5$ . Plugging these values into the formula, we get:

y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-5)}}{2(1)}

Simplifying, we get:

y = \frac{4 \pm \sqrt{16 + 20}}{2}

y = \frac{4 \pm \sqrt{36}}{2}

y = \frac{4 \pm 6}{2}

We have two possible values for $y$ :

y = \frac{4 + 6}{2} = 5

y = \frac{4 - 6}{2} = -1

Since the length of a side cannot be negative, we discard the solution $y = -1$ . Therefore, the length of the longer leg is $y = 5$ feet.

Finding the Length of the Shorter Leg

Now that we have found the length of the longer leg, we can find the length of the shorter leg using the equation:

x = y - 4

Substituting $y = 5$ , we get:

x = 5 - 4 = 1

Therefore, the length of the shorter leg is $x = 1$ foot.

Finding the Length of the Hypotenuse

We are given that the length of the hypotenuse is $\sqrt{26}$ feet. However, we can also find it using the Pythagorean theorem:

x^2 + y^2 = (\sqrt{26})^2

Substituting $x = 1$ and $y = 5$ , we get:

1^2 + 5^2 = (\sqrt{26})^2

1 + 25 = 26

This confirms that the length of the hypotenuse is indeed $\sqrt{26}$ feet.

Conclusion

In this article, we have found the lengths of the three sides of a right triangle with a specific condition. We have used the Pythagorean theorem and quadratic equations to solve the problem. The lengths of the three sides are:

Shorter leg: 1 foot
Longer leg: 5 feet
Hypotenuse: $\sqrt{26}$ feet

Introduction

In our previous article, we explored a problem involving a right triangle with a specific condition. We found the lengths of the three sides of the triangle using the Pythagorean theorem and quadratic equations. In this article, we will answer some common questions related to the problem.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) 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 problems?

A: To use the Pythagorean theorem, you need to know the lengths of two sides of a right triangle and the length of the hypotenuse. You can then use the formula:

a^2 + b^2 = c^2

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

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a squared variable. It is typically written in the form:

ax^2 + bx + c = 0

where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and graphing. The quadratic formula is a popular method that involves using the formula:

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

Q: What is the difference between a right triangle and an oblique triangle?

A: A right triangle is a triangle with one right angle (90 degrees), while an oblique triangle is a triangle with no right angles. Oblique triangles can be further classified into acute triangles (all angles are acute) and obtuse triangles (one angle is obtuse).

Q: How do I find the length of the hypotenuse of a right triangle?

A: To find the length of the hypotenuse of a right triangle, you can use the Pythagorean theorem:

a^2 + b^2 = c^2

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

Q: What is the relationship between the lengths of the sides of a right triangle?

A: The lengths of the sides of a right triangle are related by the Pythagorean theorem:

a^2 + b^2 = c^2

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

Conclusion

In this article, we have answered some common questions related to the problem of finding the lengths of the three sides of a right triangle with a specific condition. We hope this article has provided a clear and concise explanation of the concepts involved.

Additional Resources

Practice Problems

Find the length of the hypotenuse of a right triangle with legs of length 3 and 4.
Solve the quadratic equation $x^2 + 5x + 6 = 0$ .
Find the length of the shorter leg of a right triangle with a hypotenuse of length $\sqrt{13}$ and a longer leg of length 3.

We hope these practice problems help you to reinforce your understanding of the concepts involved.