Expressing the Hypotenuse in Terms of a

Oct 13, 2025 by windailys.com 17 views

Introduction

In a right triangle, the hypotenuse is the longest side, opposite the right angle. The relationship between the hypotenuse and the other two sides is governed by the Pythagorean theorem. In this article, we will explore how to solve for the value of one side of a right triangle when the hypotenuse is expressed in terms of the other side.

The Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:

c^2 = a^2 + b^2

Expressing the Hypotenuse in Terms of a

In this problem, we are given that the hypotenuse (c) is equal to a times the square root of 2 (a√2). We can express this mathematically as:

c = a√2

Substituting the Value of c into the Pythagorean Theorem

We can substitute the value of c into the Pythagorean theorem to get:

(a√2)^2 = a^2 + b^2

Simplifying the Equation

We can simplify the equation by expanding the left-hand side:

2a^2 = a^2 + b^2

Subtracting a^2 from Both Sides

We can subtract a^2 from both sides of the equation to get:

a^2 = b^2

Solving for b

We can take the square root of both sides of the equation to get:

b = a

Substituting the Value of b into the Original Equation

We can substitute the value of b into the original equation to get:

c = a√2

Substituting the Value of c

We are given that c = 6. We can substitute this value into the equation to get:

6 = a√2

Solving for a

We can solve for a by dividing both sides of the equation by √2:

a = 6 / √2

Rationalizing the Denominator

We can rationalize the denominator by multiplying both the numerator and the denominator by √2:

a = (6 / √2) × (√2 / √2)

Simplifying the Expression

We can simplify the expression by canceling out the √2 terms:

a = 6√2 / 2

Simplifying Further

We can simplify the expression further by dividing both the numerator and the denominator by 2:

a = 3√2

Conclusion

In this article, we have shown how to solve for the value of a in a right triangle when the hypotenuse is expressed in terms of the other side. We have used the Pythagorean theorem and algebraic manipulation to arrive at the solution. The final answer is:

The Value of a is 3√2

Discussion

This problem is a classic example of how to use the Pythagorean theorem to solve for the value of a side of a right triangle. The key insight is to express the hypotenuse in terms of the other side and then use algebraic manipulation to solve for the value of a. This type of problem is commonly encountered in mathematics and physics, and is an important tool for solving problems involving right triangles.

Solutions to Related Problems

If the hypotenuse is c = a√3, then the value of a is 6 / √3.
If the hypotenuse is c = a√5, then the value of a is 6 / √5.
If the hypotenuse is c = a√2, then the value of a is 3√2.

Final Thoughts

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 (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:

c^2 = a^2 + b^2

Q: How do I use the Pythagorean theorem to solve for the value of a side of a right triangle?

A: To use the Pythagorean theorem to solve for the value of a side of a right triangle, you need to know the lengths of the other two sides. You can then plug these values into the equation and solve for the unknown side.

Q: What is the difference between the Pythagorean theorem and the Pythagorean identity?

A: The Pythagorean theorem is a statement about the relationship between the lengths of the sides of a right triangle, while the Pythagorean identity is a mathematical statement that describes the relationship between the squares of the lengths of the sides of a right triangle. The Pythagorean identity is a more general statement that can be applied to any right triangle, while the Pythagorean theorem is a specific statement that applies to a particular right triangle.

Q: Can I use the Pythagorean theorem to solve for the value of the hypotenuse?

A: Yes, you can use the Pythagorean theorem to solve for the value of the hypotenuse. If you know the lengths of the other two sides, you can plug these values into the equation and solve for the hypotenuse.

Q: What is the value of a in the equation c = a√2?

A: The value of a in the equation c = a√2 is 3√2.

Q: How do I rationalize the denominator in an equation?

A: To rationalize the denominator in an equation, you need to multiply both the numerator and the denominator by the square root of the denominator. This will eliminate the square root from the denominator.

Q: What is the difference between a rationalized denominator and an irrational denominator?

A: A rationalized denominator is a denominator that has been multiplied by the square root of the denominator to eliminate the square root. An irrational denominator is a denominator that contains a square root.

Q: Can I use the Pythagorean theorem to solve for the value of a side of a right triangle if the hypotenuse is not a perfect square?

A: Yes, you can use the Pythagorean theorem to solve for the value of a side of a right triangle if the hypotenuse is not a perfect square. However, you will need to use a calculator or other mathematical tool to find the square root of the hypotenuse.

Q: What is the value of a in the equation c = a√3?

A: The value of a in the equation c = a√3 is 6 / √3.

Q: How do I simplify an expression with a square root in the denominator?

A: To simplify an expression with a square root in the denominator, you need to multiply both the numerator and the denominator by the square root of the denominator. This will eliminate the square root from the denominator.

Q: What is the difference between a simplified expression and a complex expression?

A: A simplified expression is an expression that has been reduced to its simplest form, while a complex expression is an expression that contains multiple terms or operations.

Q: Can I use the Pythagorean theorem to solve for the value of a side of a right triangle if the other two sides are not integers?

A: Yes, you can use the Pythagorean theorem to solve for the value of a side of a right triangle if the other two sides are not integers. However, you will need to use a calculator or other mathematical tool to find the square root of the hypotenuse.

Q: What is the value of a in the equation c = a√5?

A: The value of a in the equation c = a√5 is 6 / √5.

Conclusion

In this article, we have answered some of the most frequently asked questions about the Pythagorean theorem and its applications. We have covered topics such as the Pythagorean theorem, the Pythagorean identity, rationalizing the denominator, and simplifying expressions with square roots. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the Pythagorean theorem.