Applying the Pythagorean Theorem to the Pennant

Oct 13, 2025 by miodeya.com 21 views

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Introduction

In geometry, a pennant is a type of quadrilateral with two pairs of adjacent sides that are equal in length. In this problem, we are given a pennant with sides of 13, an altitude of $$x + 7$$, and a base of $$2x$$. Our goal is to find the length of the base, which is represented by the variable $$2x$$. To solve this problem, we will use the properties of a pennant and the Pythagorean theorem.

Properties of a Pennant

A pennant is a type of quadrilateral with two pairs of adjacent sides that are equal in length. This means that the opposite sides of a pennant are not necessarily equal in length. However, the two pairs of adjacent sides are equal in length, which gives us some useful information that we can use to solve this problem.

The Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that states that in a right-angled triangle, 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. This theorem can be expressed mathematically as:

a^2 + b^2 = c^2

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

Applying the Pythagorean Theorem to the Pennant

In this problem, we are given a pennant with an altitude of $$x + 7$$ and a base of $$2x$$. We can use the Pythagorean theorem to find the length of the other side of the pennant, which is equal to 13. Let's call this side $$a$$. We can then use the Pythagorean theorem to set up an equation and solve for $$x$$.

Setting Up the Equation

Using the Pythagorean theorem, we can set up the following equation:

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

Expanding the left-hand side of the equation, we get:

x^2 + 14x + 49 + 4x^2 = 169

Combine like terms:

5x^2 + 14x - 120 = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

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

In this case, $$a = 5$$, $$b = 14$$, and $$c = -120$$. Plugging these values into the quadratic formula, we get:

x = \frac{-14 \pm \sqrt{14^2 - 4(5)(-120)}}{2(5)}

Simplifying the expression under the square root, we get:

x = \frac{-14 \pm \sqrt{196 + 2400}}{10}

x = \frac{-14 \pm \sqrt{2596}}{10}

x = \frac{-14 \pm 51.04}{10}

Finding the Length of the Base

Now that we have found the value of $$x$$, we can find the length of the base by plugging this value into the expression $$2x$$. We get:

2x = 2(5.104)

2x = 10.208

Conclusion

In this problem, we were given a pennant with sides of 13, an altitude of $$x + 7$$, and a base of $$2x$$. We used the Pythagorean theorem to set up an equation and solve for $$x$$. We then used this value to find the length of the base, which is $$10.208$$. This problem demonstrates the importance of using geometric properties and the Pythagorean theorem to solve problems involving variable dimensions.

Final Answer

The final answer is: $\boxed{10.208}$

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Introduction

In our previous article, we explored the problem of finding the length of the base of a pennant with variable dimensions. We used the Pythagorean theorem to set up an equation and solve for the variable $$x$$. In this article, we will answer some common questions that readers may have about this problem.

Q: What is a pennant in geometry?

A: A pennant is a type of quadrilateral with two pairs of adjacent sides that are equal in length. This means that the opposite sides of a pennant are not necessarily equal in length, but the two pairs of adjacent sides are equal in length.

Q: How do I know which sides are equal in length?

A: To determine which sides are equal in length, you can use the properties of a pennant. Since a pennant has two pairs of adjacent sides that are equal in length, you can identify these sides by looking for pairs of sides that are adjacent to each other.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right-angled triangle, 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. This theorem can be expressed mathematically as:

a^2 + b^2 = c^2

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

Q: How do I apply the Pythagorean theorem to a pennant?

A: To apply the Pythagorean theorem to a pennant, you need to identify the right-angled triangle within the pennant. In this problem, we used the altitude of the pennant as one of the sides of the right-angled triangle, and the base of the pennant as the other side. We then used the Pythagorean theorem to set up an equation and solve for the variable $$x$$.

Q: What is the significance of the variable $$x$$ in this problem?

A: The variable $$x$$ represents the length of the base of the pennant. By solving for $$x$$, we were able to find the length of the base, which is $$10.208$$.

Q: Can I use the Pythagorean theorem to solve other problems involving pennants?

A: Yes, you can use the Pythagorean theorem to solve other problems involving pennants. The Pythagorean theorem is a general concept that can be applied to any right-angled triangle, including those that are part of a pennant.

Q: What are some common mistakes to avoid when solving problems involving pennants?

A: Some common mistakes to avoid when solving problems involving pennants include:

Failing to identify the right-angled triangle within the pennant
Failing to apply the Pythagorean theorem correctly
Failing to solve for the variable $$x$$ correctly

Q: How can I practice solving problems involving pennants?

A: You can practice solving problems involving pennants by working through example problems and exercises. You can also try creating your own problems involving pennants and solving them using the Pythagorean theorem.

Conclusion

In this article, we answered some common questions that readers may have about the problem of finding the length of the base of a pennant with variable dimensions. We also provided some tips and advice for practicing solving problems involving pennants. By following these tips and practicing regularly, you can become more confident and proficient in solving problems involving pennants.

Final Answer

The final answer is: $\boxed{10.208}$