Finding The Height Of Equilateral Triangle MNO Side $16\sqrt{3}$

Before we dive into solving the problem, let's first understand the fascinating world of equilateral triangles. An equilateral triangle is a triangle with three equal sides and three equal angles. Each angle in an equilateral triangle measures 60 degrees. This unique property makes them not only visually appealing but also mathematically significant. The symmetry inherent in equilateral triangles leads to several interesting characteristics, including the relationship between their sides and their heights.

One of the key features of an equilateral triangle is that the altitude, which is the perpendicular line drawn from a vertex to the opposite side, also acts as the median and the angle bisector. This means that the altitude divides the base into two equal segments and the vertex angle into two equal angles. This property is crucial when calculating the height of the triangle, as it creates two congruent right-angled triangles within the equilateral triangle. Understanding this symmetry simplifies the process of finding the height, allowing us to leverage trigonometric ratios or the Pythagorean theorem effectively. The height, in essence, bisects the equilateral triangle, forming two 30-60-90 right triangles, a special type of triangle with well-defined side ratios.

Furthermore, the relationship between the side length and the height of an equilateral triangle is constant. This relationship can be derived using the Pythagorean theorem or trigonometric ratios, making it a fundamental concept in geometry. Knowing this relationship allows for quick calculations and problem-solving, particularly in scenarios where only the side length or the height is given. The consistent proportions within equilateral triangles make them predictable and easy to work with, which is why they often appear in mathematical problems and real-world applications.

The exploration of equilateral triangles also extends to their area calculation. The area of an equilateral triangle can be calculated using a formula that involves the side length, further highlighting the importance of understanding the properties of these shapes. The interplay between the sides, angles, height, and area of an equilateral triangle makes it a cornerstone in the study of geometry. Understanding these relationships is not only crucial for solving mathematical problems but also for appreciating the geometric harmony found in nature and architecture.

Now, let's tackle the specific problem at hand. We have an equilateral triangle MNO with sides measuring $16\sqrt{3}$ units. Our mission is to determine the height of this triangle. This problem serves as an excellent example of how the properties of equilateral triangles can be applied to solve real mathematical challenges. The given side length provides us with the key information needed to unlock the height, and by utilizing the principles we've discussed, we can arrive at the correct answer. The process involves breaking down the equilateral triangle into manageable parts and applying relevant geometric theorems.

The first step in solving this problem is visualizing the triangle and the height. Imagine drawing a perpendicular line from vertex M to side NO. This line represents the height of the triangle, and it also divides the equilateral triangle into two congruent right-angled triangles. Each of these right-angled triangles has a hypotenuse equal to the side length of the equilateral triangle ($16\sqrt{3}$ units), a base equal to half the side length, and a height that we need to find. This visual representation helps to clarify the relationships between the sides and angles, making it easier to apply the necessary formulas or theorems.

The height not only bisects the base but also forms a 30-60-90 triangle. This is a special right triangle with well-defined side ratios. The sides are in the ratio of 1:$\sqrt{3}$:2, where the shortest side is opposite the 30-degree angle, the middle side is opposite the 60-degree angle, and the longest side (the hypotenuse) is opposite the 90-degree angle. Recognizing this special triangle allows us to use these ratios to find the height without having to rely solely on the Pythagorean theorem. The 30-60-90 triangle is a powerful tool in geometry, and its application here demonstrates its versatility in problem-solving.

To find the height of the equilateral triangle MNO, we can leverage the properties of 30-60-90 triangles. As mentioned earlier, the height divides the equilateral triangle into two congruent 30-60-90 triangles. The hypotenuse of each of these triangles is $16\sqrt{3}$ units, which is the side length of the equilateral triangle. The base of each triangle is half the side length of the equilateral triangle, which is $\frac{16\sqrt{3}}{2} = 8\sqrt{3}$ units. Now, we need to find the height, which is the side opposite the 60-degree angle in the 30-60-90 triangle.

In a 30-60-90 triangle, the ratio of the sides opposite the 30-degree, 60-degree, and 90-degree angles is 1:$\sqrt{3}$:2. Let's denote the side opposite the 30-degree angle as x, the side opposite the 60-degree angle (the height) as $x\sqrt{3}$, and the side opposite the 90-degree angle (the hypotenuse) as 2x. We know that the hypotenuse is $16\sqrt{3}$ units, so we can set up the equation 2x = $16\sqrt{3}$. Solving for x, we get x = $8\sqrt{3}$.

Now that we have the value of x, we can find the height, which is $x\sqrt{3}$. Substituting x = $8\sqrt{3}$, we get height = $(8\sqrt{3})\sqrt{3} = 8 * 3 = 24$ units. Therefore, the height of the equilateral triangle MNO is 24 units. This methodical approach, utilizing the 30-60-90 triangle properties, allows for a clear and concise solution. The ability to recognize and apply these special triangle ratios is a valuable skill in geometry, enabling efficient problem-solving.

Alternatively, we can find the height of the triangle using the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In our case, the hypotenuse is $16\sqrt{3}$ units, one side is half the base ($8\sqrt{3}$ units), and the other side is the height (which we'll call h).

Applying the Pythagorean theorem, we have: $(16\sqrt{3})^2 = (8\sqrt{3})^2 + h^2$. Simplifying the equation, we get: $16^2 * 3 = 8^2 * 3 + h^2$, which is $768 = 192 + h^2$. Subtracting 192 from both sides, we get $h^2 = 576$. Taking the square root of both sides, we find h = 24 units. This confirms our previous solution using the 30-60-90 triangle properties.

The use of the Pythagorean theorem provides an independent verification of our result, reinforcing the accuracy of our calculations. This approach highlights the versatility of geometric principles and the ability to solve the same problem using different methods. The Pythagorean theorem is a fundamental concept in geometry, and its application here demonstrates its broad applicability in solving problems involving right-angled triangles. This alternative solution not only provides a different perspective but also enhances our understanding of the geometric relationships within the triangle.

In conclusion, the height of the equilateral triangle MNO with sides measuring $16\sqrt{3}$ units is 24 units. We arrived at this answer using two different methods: the properties of 30-60-90 triangles and the Pythagorean theorem. Both methods provide a clear and concise solution, demonstrating the interconnectedness of geometric concepts. Understanding the properties of equilateral triangles and being able to apply them in problem-solving scenarios is a crucial skill in mathematics.

This problem exemplifies the beauty and elegance of geometry. The ability to break down a complex shape into simpler components and apply fundamental principles is a testament to the power of mathematical reasoning. The two solutions presented not only provide the correct answer but also illustrate different approaches to problem-solving, enriching our understanding of the subject. The consistent result obtained through both methods reinforces the validity of the solution and highlights the robustness of geometric principles. This exercise underscores the importance of having a strong foundation in geometric concepts and the ability to apply them flexibly in various contexts.

The correct answer is B. 24 units.