Triangle QRS Find The Hypotenuse Length

In this article, we will delve into the properties of triangle QRS, specifically focusing on calculating the length of its hypotenuse. We are given that the legs of triangle QRS each have a length of 10 units. To determine the hypotenuse, we need to understand the type of triangle QRS is and apply the appropriate mathematical principles.

Identifying the Triangle Type

When analyzing a triangle, the first step is to identify its type. Knowing the characteristics of the triangle will dictate which formulas and theorems we can apply. In this case, we are told that the legs of triangle QRS each have a length of 10 units. This key piece of information indicates that the triangle is an isosceles right triangle. Let's break down what this means:

• Isosceles Triangle: An isosceles triangle is a triangle with at least two sides of equal length. In our scenario, since both legs of triangle QRS have a length of 10 units, it satisfies the condition for being an isosceles triangle.
• Right Triangle: A right triangle is a triangle that has one angle measuring 90 degrees. This angle is often referred to as the right angle. In a right triangle, the two sides that form the right angle are called legs, and the side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side of a right triangle.

Given that triangle QRS has two legs of equal length and we are implicitly working within the context of finding a hypotenuse (which only exists in right triangles), we can confidently classify triangle QRS as an isosceles right triangle. This classification is crucial because it allows us to use specific properties and theorems associated with this type of triangle.

Applying the Pythagorean Theorem

Now that we have identified triangle QRS as an isosceles right triangle, we can use the Pythagorean Theorem to calculate the length of the hypotenuse. The Pythagorean Theorem is a fundamental concept in geometry that relates 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 (the legs). Mathematically, this is expressed as:

$$a^2 + b^2 = c^2$$

Where:

• a and b are the lengths of the legs of the right triangle.
• c is the length of the hypotenuse.

In our case, we know that the legs of triangle QRS each have a length of 10 units. Let's denote these legs as a and b, so a = 10 and b = 10. We want to find the length of the hypotenuse, which we'll denote as c. Plugging the values of a and b into the Pythagorean Theorem, we get:

$$10^2 + 10^2 = c^2$$

Now, we can solve for c:

$$100 + 100 = c^2$$

$$200 = c^2$$

To find c, we take the square root of both sides of the equation:

$$c = \sqrt{200}$$

Simplifying the Result

We have found that the length of the hypotenuse is the square root of 200. To simplify this, we can factor 200 into its prime factors or look for perfect square factors. We can express 200 as 100 times 2, where 100 is a perfect square (10 * 10 = 100). Therefore, we can rewrite the square root of 200 as:

$$c = \sqrt{100 \times 2}$$

Using the property of square roots that states $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can separate the square root:

$$c = \sqrt{100} \times \sqrt{2}$$

Since the square root of 100 is 10, we get:

$$c = 10 \sqrt{2}$$

So, the length of the hypotenuse of triangle QRS is $10 \sqrt{2}$ units.

Conclusion

By identifying triangle QRS as an isosceles right triangle and applying the Pythagorean Theorem, we have successfully calculated the length of its hypotenuse. The hypotenuse of triangle QRS is $10 \sqrt{2}$ units. This exercise demonstrates the importance of recognizing triangle types and utilizing appropriate mathematical principles to solve geometric problems. Understanding the Pythagorean Theorem and its applications is crucial in various fields, including geometry, trigonometry, and even practical applications in construction and engineering.

In this comprehensive review, we will revisit the key concepts involved in determining the length of the hypotenuse of triangle QRS. Triangle QRS is defined as having legs each with a length of 10 units. Our goal is to reinforce the understanding of triangle types, the application of the Pythagorean Theorem, and the simplification of radical expressions. This review will ensure that the process of finding the hypotenuse length is clear and well-understood.

Recapping Triangle Identification

To accurately calculate the hypotenuse, it is essential to correctly identify the type of triangle we are dealing with. The given information states that triangle QRS has legs of equal length, specifically 10 units each. This immediately suggests that the triangle is an isosceles triangle. Recall that an isosceles triangle is defined as a triangle with at least two sides of equal length. In the case of triangle QRS, the two legs are of equal length, fulfilling the criteria for an isosceles triangle.

Additionally, the problem implies that triangle QRS is a right triangle. This is because the concept of a hypotenuse is exclusively associated with right triangles. A right triangle is characterized by having one angle that measures 90 degrees, known as the right angle. The side opposite the right angle is the hypotenuse, and the two sides forming the right angle are the legs. Since we are asked to find the hypotenuse, we can infer that triangle QRS is indeed a right triangle.

Combining these two properties, we conclude that triangle QRS is an isosceles right triangle. This classification is vital because it allows us to leverage the specific properties and theorems associated with this type of triangle, particularly the Pythagorean Theorem.

Step-by-Step Application of the Pythagorean Theorem

The Pythagorean Theorem is the cornerstone for calculating the hypotenuse of a right triangle. This theorem establishes a relationship between the lengths of the sides of a right triangle, stating 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 (the legs). The mathematical representation of the Pythagorean Theorem is:

$$a^2 + b^2 = c^2$$

Where:

• a and b are the lengths of the legs of the right triangle.
• c is the length of the hypotenuse.

In the context of triangle QRS, we are given that the legs each have a length of 10 units. Let's assign these lengths to a and b, so a = 10 and b = 10. Our objective is to determine the length of the hypotenuse, c. Substituting the known values into the Pythagorean Theorem, we get:

$$10^2 + 10^2 = c^2$$

Let's break down the calculation step-by-step:

1. Square the lengths of the legs:

   $$10^2 = 10 \times 10 = 100$$

   So, we have:

   $$100 + 100 = c^2$$

2. Add the squared values:

   $$100 + 100 = 200$$

   Thus, the equation becomes:

   $$200 = c^2$$

3. Find the square root of both sides:

   To isolate c, we take the square root of both sides of the equation:

   $$c = \sqrt{200}$$

Simplifying the Square Root of 200

We have found that the length of the hypotenuse, c, is the square root of 200. However, this value can be simplified to a more manageable form. To simplify a square root, we look for perfect square factors within the radicand (the number under the square root symbol). In this case, 200 can be factored into 100 and 2, where 100 is a perfect square (10 * 10 = 100). We can rewrite the square root of 200 as:

$$c = \sqrt{100 \times 2}$$

Using the property of square roots that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can separate the square root:

$$c = \sqrt{100} \times \sqrt{2}$$

Now, we know that the square root of 100 is 10:

$$c = 10 \times \sqrt{2}$$

Therefore, the simplified form of the hypotenuse length is:

$$c = 10 \sqrt{2}$$

Final Hypotenuse Length

Through the application of the Pythagorean Theorem and the simplification of the resulting square root, we have determined that the length of the hypotenuse of triangle QRS is $10 \sqrt{2}$ units. This final result confirms the length of the hypotenuse in its simplest form, making it easier to understand and work with in further calculations or applications.

Conclusion: Mastering Hypotenuse Calculation

In conclusion, this review has reinforced the key steps involved in calculating the length of the hypotenuse of triangle QRS. By accurately identifying the triangle as an isosceles right triangle and applying the Pythagorean Theorem, we were able to determine that the hypotenuse length is $10 \sqrt{2}$ units. The process included squaring the lengths of the legs, summing the squared values, finding the square root of the sum, and simplifying the resulting radical expression. A solid understanding of these concepts is crucial for solving various geometric problems and real-world applications involving right triangles.

Analyzing the Answer Choices

Now, let's consider the multiple-choice options provided:

• A. 5 units
• B. $5 \sqrt{2}$ units
• C. 10 units
• D. $10 \sqrt{2}$ units

Based on our calculations, the correct answer is D. $10 \sqrt{2}$ units.

This comprehensive analysis of triangle QRS, from identifying its type to calculating and simplifying the hypotenuse length, provides a solid understanding of the concepts involved. The step-by-step approach ensures clarity and accuracy in solving geometric problems of this nature. The correct option emphasizes the importance of understanding geometric principles and their practical applications.