The Hypotenuse Of An Isosceles Triangle Measures 10 Inches. What Is The Length Of One Leg Of The Triangle?A. $\frac{10}{\sqrt{3}}$B. $\frac{10}{\sqrt{2}}$C. $10 \sqrt{2}$D. $10 \sqrt{3}$
Introduction
In the realm of geometry, triangles are a fundamental concept that has been studied and analyzed for centuries. Among the various types of triangles, the isosceles triangle stands out for its unique properties. An isosceles triangle is a triangle with two sides of equal length, which are called the legs. The third side, opposite the base angles, is called the hypotenuse. In this article, we will delve into the world of isosceles triangles and explore the relationship between the hypotenuse and the legs. Specifically, we will examine the problem of finding the length of one leg of an isosceles triangle when the hypotenuse measures 10 inches.
Understanding Isosceles Triangles
An isosceles triangle has two sides of equal length, which are the legs. The third side, the hypotenuse, is opposite the base angles. Since the two legs are equal in length, the triangle is symmetrical about the altitude from the vertex angle to the base. This symmetry is a key characteristic of isosceles triangles and plays a crucial role in solving problems involving these triangles.
The Pythagorean Theorem
The Pythagorean Theorem is a fundamental concept in geometry that relates the lengths of the sides of a right triangle. The 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:
a^2 + b^2 = c^2
In the case of an isosceles triangle, since the two legs are equal in length, we can let a = b. Substituting this into the Pythagorean Theorem, we get:
a^2 + a^2 = c^2 2a^2 = c^2
Applying the Pythagorean Theorem to the Problem
Now that we have a deeper understanding of isosceles triangles and the Pythagorean Theorem, we can apply this knowledge to the problem at hand. We are given that the hypotenuse of an isosceles triangle measures 10 inches. Using the Pythagorean Theorem, we can set up an equation to solve for the length of one leg:
2a^2 = 10^2 2a^2 = 100
Solving for the Length of One Leg
To solve for the length of one leg, we need to isolate the variable a. We can do this by dividing both sides of the equation by 2:
a^2 = 100/2 a^2 = 50
Taking the Square Root
To find the length of one leg, we need to take the square root of both sides of the equation:
a = √50 a = √(25 × 2) a = √25 × √2 a = 5√2
Conclusion
In conclusion, we have successfully applied the Pythagorean Theorem to find the length of one leg of an isosceles triangle when the hypotenuse measures 10 inches. The length of one leg is given by the expression 5√2. This result is consistent with option C, which states that the length of one leg is 10√2. However, we must note that the correct answer is 5√2, not 10√2.
Final Answer
The final answer is .
Discussion
The problem presented in this article is a classic example of how the Pythagorean Theorem can be applied to solve problems involving right triangles. The symmetry of isosceles triangles makes them particularly well-suited for this type of problem. By understanding the properties of isosceles triangles and the Pythagorean Theorem, we can solve a wide range of problems involving these triangles.
Related Problems
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 15 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 20 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 25 inches.
Solved Problems
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 10 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 12 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 15 inches.
Practice Problems
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 18 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 22 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 25 inches.
Conclusion
In conclusion, the Pythagorean Theorem is a powerful tool for solving problems involving right triangles. By understanding the properties of isosceles triangles and the Pythagorean Theorem, we can solve a wide range of problems involving these triangles. The problem presented in this article is a classic example of how the Pythagorean Theorem can be applied to solve problems involving right triangles.
Introduction
In our previous article, we explored the relationship between the hypotenuse and the legs of an isosceles triangle. We applied the Pythagorean Theorem to find the length of one leg when the hypotenuse measures 10 inches. In this article, we will answer some of the most frequently asked questions related to the hypotenuse of an isosceles triangle.
Q1: What is the relationship between the hypotenuse and the legs of an isosceles triangle?
A1: The hypotenuse of an isosceles triangle is the side opposite the base angles. Since the two legs are equal in length, the triangle is symmetrical about the altitude from the vertex angle to the base.
Q2: How do I find the length of one leg of an isosceles triangle when the hypotenuse is given?
A2: To find the length of one leg of an isosceles triangle when the hypotenuse is given, you can use the Pythagorean Theorem. Let a be the length of one leg and c be the length of the hypotenuse. Then, a^2 + a^2 = c^2. Simplifying this equation, we get 2a^2 = c^2. Solving for a, we get a = √(c^2/2).
Q3: What is the formula for finding the length of one leg of an isosceles triangle?
A3: The formula for finding the length of one leg of an isosceles triangle is a = √(c^2/2), where a is the length of one leg and c is the length of the hypotenuse.
Q4: Can I use the Pythagorean Theorem to find the length of one leg of an isosceles triangle when the hypotenuse is not given?
A4: No, you cannot use the Pythagorean Theorem to find the length of one leg of an isosceles triangle when the hypotenuse is not given. The Pythagorean Theorem requires the length of the hypotenuse to be known.
Q5: What is the relationship between the hypotenuse and the base of an isosceles triangle?
A5: The hypotenuse of an isosceles triangle is the side opposite the base angles. The base of an isosceles triangle is the side that is not equal to the other two sides.
Q6: How do I find the length of the base of an isosceles triangle when the hypotenuse is given?
A6: To find the length of the base of an isosceles triangle when the hypotenuse is given, you can use the Pythagorean Theorem. Let b be the length of the base and c be the length of the hypotenuse. Then, b^2 + a^2 = c^2, where a is the length of one leg. Since the two legs are equal in length, we can let a = b. Substituting this into the equation, we get 2b^2 = c^2. Solving for b, we get b = √(c^2/2).
Q7: What is the formula for finding the length of the base of an isosceles triangle?
A7: The formula for finding the length of the base of an isosceles triangle is b = √(c^2/2), where b is the length of the base and c is the length of the hypotenuse.
Q8: Can I use the Pythagorean Theorem to find the length of the base of an isosceles triangle when the hypotenuse is not given?
A8: No, you cannot use the Pythagorean Theorem to find the length of the base of an isosceles triangle when the hypotenuse is not given. The Pythagorean Theorem requires the length of the hypotenuse to be known.
Conclusion
In conclusion, the Pythagorean Theorem is a powerful tool for solving problems involving right triangles. By understanding the properties of isosceles triangles and the Pythagorean Theorem, we can solve a wide range of problems involving these triangles. The questions and answers presented in this article provide a comprehensive overview of the relationship between the hypotenuse and the legs of an isosceles triangle.
Final Answer
The final answer is .
Related Problems
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 15 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 20 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 25 inches.
Solved Problems
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 10 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 12 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 15 inches.
Practice Problems
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 18 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 22 inches.
- Find the length of one leg of an isosceles triangle when the hypotenuse measures 25 inches.