Altitude Length Of Isosceles Right Triangle With 4 Cm Legs

Jul 16, 2025 by Jeany 59 views

Isosceles Right Triangle Altitude Length Calculation

This article delves into the fascinating world of geometry, specifically focusing on the properties of isosceles right triangles. We will explore how to determine the length of the altitude drawn from the right angle to the hypotenuse in such a triangle. This problem is not only a classic example in geometry but also a fundamental concept that bridges various mathematical principles. Understanding this concept is crucial for students and enthusiasts alike, as it lays the groundwork for more advanced geometric problem-solving. The specific problem we will tackle involves an isosceles right triangle with leg lengths of 4 centimeters. Our goal is to find the length of the altitude drawn from the right angle to the hypotenuse. This seemingly simple problem involves a combination of geometric theorems and algebraic manipulation, providing a rich learning experience. By the end of this article, you will have a clear understanding of how to approach this type of problem and the underlying principles that govern it. We will break down the problem step by step, ensuring that each concept is thoroughly explained and easy to grasp. From the Pythagorean theorem to the properties of similar triangles, we will cover all the necessary tools to solve this problem effectively. So, let's embark on this geometric journey and uncover the solution together!

Problem Statement

We are given an isosceles right triangle with leg lengths of 4 centimeters. The main objective is to determine the length of the altitude drawn from the right angle to the hypotenuse. Before diving into the solution, it's essential to understand the key terms and concepts involved. An isosceles right triangle, also known as a 45-45-90 triangle, is a triangle with two equal sides (legs) and one right angle (90 degrees). The side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle. The altitude, in this context, is a line segment drawn from the right angle perpendicular to the hypotenuse. This altitude divides the original triangle into two smaller triangles, each of which is also an isosceles right triangle. Understanding these basic properties is crucial for solving the problem at hand. The problem requires us to find the length of this altitude. This involves using several geometric principles, such as the Pythagorean theorem, the properties of similar triangles, and the relationships between sides and angles in an isosceles right triangle. To tackle this problem effectively, we will first determine the length of the hypotenuse using the Pythagorean theorem. Then, we will use the concept of similar triangles and the properties of altitudes to find the length of the altitude. This step-by-step approach will ensure clarity and a thorough understanding of the solution. Let's proceed with the solution, keeping these concepts in mind.

Solution

To solve this problem, we will employ a step-by-step approach, combining the Pythagorean theorem and the properties of similar triangles. First, let's label the triangle ABC, where angle C is the right angle, and sides AC and BC are the legs with lengths of 4 cm each. The side AB is the hypotenuse, and we need to find the length of the altitude CD drawn from C to AB.

Step 1: Find the Length of the Hypotenuse

Using the Pythagorean theorem, which 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), we have:

$AB^2 = AC^2 + BC^2$

Given that AC = 4 cm and BC = 4 cm, we substitute these values into the equation:

$AB^2 = 4^2 + 4^2$ $AB^2 = 16 + 16$ $AB^2 = 32$

Taking the square root of both sides, we get:

$AB = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$ cm

So, the length of the hypotenuse AB is $4\sqrt{2}$ cm.

Step 2: Understand the Properties of the Altitude

The altitude CD divides the isosceles right triangle ABC into two smaller triangles, ADC and BDC. Since ABC is an isosceles right triangle, angles A and B are both 45 degrees. The altitude CD is perpendicular to AB, which means angles CDA and CDB are both 90 degrees. This creates two smaller right triangles that are also isosceles right triangles.

Step 3: Use the Area of the Triangle

There are two ways to calculate the area of triangle ABC:

Using the legs as the base and height:

Area = $\frac{1}{2} \times AC \times BC = \frac{1}{2} \times 4 \times 4 = 8$ $cm^2$
Using the hypotenuse as the base and the altitude CD as the height:

Area = $\frac{1}{2} \times AB \times CD$

Since the area is the same regardless of the method used, we can set the two expressions equal to each other:

$\frac{1}{2} \times AB \times CD = 8$

Step 4: Solve for the Length of the Altitude CD

Substitute the value of AB we found earlier ( $4\sqrt{2}$ cm) into the equation:

$\frac{1}{2} \times 4\sqrt{2} \times CD = 8$ $2\sqrt{2} \times CD = 8$

Now, solve for CD:

$CD = \frac{8}{2\sqrt{2}} = \frac{4}{\sqrt{2}}$

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$ :

$CD = \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$ cm

Therefore, the length of the altitude drawn from the right angle to the hypotenuse is $2\sqrt{2}$ cm.

Answer

The length of the altitude drawn from the right angle to the hypotenuse is B. $2 \sqrt{2}$ cm.

Detailed Explanation of Key Concepts

1. Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the sides of a right triangle. It states that in a right 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 (legs). This can be expressed mathematically as:

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

where a and b are the lengths of the legs, and c is the length of the hypotenuse.

Application in the Problem

In our problem, we used the Pythagorean theorem to find the length of the hypotenuse of the isosceles right triangle. Given that the legs had lengths of 4 cm each, we applied the theorem as follows:

$AB^2 = AC^2 + BC^2$ $AB^2 = 4^2 + 4^2$ $AB^2 = 16 + 16$ $AB^2 = 32$ $AB = \sqrt{32} = 4\sqrt{2}$ cm

This step was crucial because the length of the hypotenuse is needed to calculate the area of the triangle and subsequently find the length of the altitude.

2. Properties of Isosceles Right Triangles

An isosceles right triangle, also known as a 45-45-90 triangle, has several unique properties that are essential for solving geometric problems. These properties include:

Two sides (legs) are of equal length.
One angle is a right angle (90 degrees), and the other two angles are each 45 degrees.
The ratio of the sides is $1:1:\sqrt{2}$ , where the legs are in the ratio of 1:1, and the hypotenuse is $\sqrt{2}$ times the length of each leg.

Application in the Problem

In our problem, recognizing that the given triangle is an isosceles right triangle allowed us to infer that the two acute angles are 45 degrees each. This understanding helped in visualizing the symmetry of the triangle and how the altitude would divide it into two smaller congruent triangles. Additionally, knowing the side ratio helped in verifying the length of the hypotenuse we calculated using the Pythagorean theorem.

3. Area of a Triangle

The area of a triangle can be calculated in several ways, depending on the information available. The most common formula is:

Area = $\frac{1}{2} \times base \times height$

where the base is any side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.

Application in the Problem

We used the concept of the area of a triangle in two different ways to solve for the altitude. First, we calculated the area using the legs as the base and height:

Area = $\frac{1}{2} \times AC \times BC = \frac{1}{2} \times 4 \times 4 = 8$ $cm^2$

Then, we calculated the area using the hypotenuse as the base and the altitude as the height:

Area = $\frac{1}{2} \times AB \times CD$

By equating these two expressions for the area, we were able to solve for the length of the altitude CD:

$\frac{1}{2} \times 4\sqrt{2} \times CD = 8$

This approach provided a direct way to find the altitude by relating it to the known area of the triangle.

4. Rationalizing the Denominator

Rationalizing the denominator is a process of eliminating any radical expressions from the denominator of a fraction. This is typically done by multiplying both the numerator and the denominator by the radical in the denominator. For example, to rationalize the denominator of $\frac{a}{\sqrt{b}}$ , we multiply both the numerator and the denominator by $\sqrt{b}$ :

$\frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$

Application in the Problem

In our problem, we encountered the fraction $\frac{4}{\sqrt{2}}$ when solving for the altitude. To rationalize the denominator, we multiplied both the numerator and the denominator by $\sqrt{2}$ :

$\frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$ cm

This step simplifies the expression and presents the answer in a more conventional form.

Conclusion

In summary, we have successfully determined the length of the altitude drawn from the right angle to the hypotenuse in an isosceles right triangle with leg lengths of 4 centimeters. By applying the Pythagorean theorem, understanding the properties of isosceles right triangles, using the area of a triangle in two different ways, and rationalizing the denominator, we found that the altitude's length is $2\sqrt{2}$ cm.

This problem highlights the interconnectedness of various geometric concepts and the importance of a step-by-step approach in problem-solving. The Pythagorean theorem allowed us to find the length of the hypotenuse, while the properties of isosceles right triangles helped us understand the relationships between the sides and angles. The area of the triangle provided a crucial link between the legs and the altitude, enabling us to set up an equation to solve for the unknown altitude. Finally, rationalizing the denominator ensured that our answer was in its simplest form.

This exercise not only reinforces our understanding of these geometric principles but also demonstrates how they can be applied in tandem to solve complex problems. Mastering these concepts is essential for further studies in mathematics and related fields. By carefully analyzing each step and understanding the underlying principles, we can confidently tackle similar problems in the future. Geometry is not just about formulas and theorems; it's about understanding spatial relationships and applying logical reasoning. This problem serves as a perfect example of how these elements come together to provide elegant solutions.

This problem and its solution can be a stepping stone for exploring more advanced topics in geometry, such as trigonometry, coordinate geometry, and three-dimensional geometry. Each concept builds upon the foundational knowledge we have discussed here. As we continue to explore mathematics, we will find that these basic principles are the building blocks for more complex ideas. Therefore, a strong grasp of these fundamentals is crucial for success in mathematics and related disciplines.