Finding The Height Of Equilateral Triangle MNO A Step-by-Step Solution

In the realm of geometry, equilateral triangles hold a special place due to their symmetrical nature and unique properties. An equilateral triangle, by definition, is a triangle with all three sides of equal length and all three angles equal to 60 degrees. This harmonious symmetry lends itself to various interesting calculations and relationships, one of which is determining the height of the triangle.

Understanding Equilateral Triangles

Before we delve into the specifics of calculating the height of our given equilateral triangle MNO, let's solidify our understanding of these geometric marvels. Equilateral triangles are characterized by their three equal sides and three equal angles, each measuring 60 degrees. This inherent symmetry leads to several important properties:

• All three medians (lines from a vertex to the midpoint of the opposite side) are equal in length.
• All three altitudes (perpendicular lines from a vertex to the opposite side) are equal in length.
• All three angle bisectors (lines dividing an angle into two equal angles) are equal in length.
• The centroid (intersection of medians), orthocenter (intersection of altitudes), incenter (intersection of angle bisectors), and circumcenter (center of the circumscribed circle) all coincide at the same point.

These properties make equilateral triangles exceptionally predictable and easy to work with in geometric calculations. The height, in particular, plays a crucial role in determining the area and other key characteristics of the triangle.

Problem Statement: Triangle MNO

Our specific challenge involves an equilateral triangle, aptly named MNO. We are given that each side of this triangle measures $16\sqrt{3}$ units. The task at hand is to determine the height of this triangle. To achieve this, we'll need to leverage our understanding of equilateral triangles and the Pythagorean theorem.

Applying the Pythagorean Theorem

To find the height of triangle MNO, we can draw an altitude from one vertex (let's say M) to the opposite side (NO). This altitude will bisect the base NO, creating two right-angled triangles. Let's call the point where the altitude intersects NO as point P. Now, we have a right-angled triangle, say MPO, where:

• MO is the hypotenuse (one of the sides of the equilateral triangle), with a length of $16\sqrt{3}$ units.
• PO is half the length of NO (since the altitude bisects the base), which is $(16\sqrt{3})/2 = 8\sqrt{3}$ units.
• MP is the height we want to find.

Now, we can apply the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In our case, this translates to:

$MO^2 = MP^2 + PO^2$

Substituting the values we know:

$(16\sqrt{3})^2 = MP^2 + (8\sqrt{3})^2$

Solving for the Height

Now, let's simplify and solve for MP, which represents the height:

$16^2 * (\sqrt{3})^2 = MP^2 + 8^2 * (\sqrt{3})^2$

$256 * 3 = MP^2 + 64 * 3$

$768 = MP^2 + 192$

Subtract 192 from both sides:

$MP^2 = 768 - 192$

$MP^2 = 576$

Now, take the square root of both sides:

$MP = \sqrt{576}$

$MP = 24$

Therefore, the height of the equilateral triangle MNO is 24 units.

Conclusion

By applying the properties of equilateral triangles and the Pythagorean theorem, we successfully determined that the height of triangle MNO is 24 units. This problem highlights the elegance and interconnectedness of geometric concepts, demonstrating how fundamental principles can be used to solve seemingly complex problems. The key takeaway is the ability to break down a problem into smaller, manageable parts and apply the appropriate theorems and formulas. In this case, understanding the properties of equilateral triangles and the Pythagorean theorem was crucial to arriving at the correct solution. This exercise not only reinforces our understanding of geometry but also hones our problem-solving skills, which are invaluable in various mathematical and real-world contexts.

Delving further into the solution for the height of equilateral triangle MNO, we can appreciate the intricacies and the logical steps involved. As a quick recap, we are dealing with an equilateral triangle where each side measures $16\sqrt{3}$ units, and our objective is to find the altitude, which represents the height of the triangle. We've already established the foundation using the Pythagorean theorem, but let's elaborate on each step for a clearer understanding.

Visualizing the Problem

The first and foremost step in solving any geometry problem is to visualize it. Imagine an equilateral triangle MNO, with all three sides equal in length. Now, visualize a line drawn from vertex M perpendicular to the opposite side NO. This line is the altitude, and it also acts as a median, bisecting the base NO into two equal segments. Let's call the point where the altitude intersects NO as point P. This visualization gives us two congruent right-angled triangles, MPO and MNP.

Identifying the Right-Angled Triangle

Focusing on one of the right-angled triangles, say MPO, we can identify the following:

• MO is the hypotenuse, which is also a side of the equilateral triangle, and its length is given as $16\sqrt{3}$ units.
• PO is half the length of NO, as the altitude MP bisects the base. Therefore, PO = $(16\sqrt{3})/2 = 8\sqrt{3}$ units.
• MP is the height we are trying to find.

This step is crucial because it sets the stage for applying the Pythagorean theorem. By identifying the right-angled triangle and its sides, we can relate the lengths of the sides using the theorem.

Applying the Pythagorean Theorem: A Step-by-Step Breakdown

The Pythagorean theorem, a cornerstone of Euclidean geometry, 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. Mathematically, this is represented as:

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

where c is the hypotenuse, and a and b are the other two sides.

In our case, applying the theorem to triangle MPO, we have:

$MO^2 = MP^2 + PO^2$

Now, let's substitute the known values:

$(16\sqrt{3})^2 = MP^2 + (8\sqrt{3})^2$

This is where the arithmetic comes into play. We need to carefully square the terms involving the square root.

Step-by-Step Calculation

Let's break down the calculation step by step:

1. $(16\sqrt{3})^2 = 16^2 * (\sqrt{3})^2 = 256 * 3 = 768$
2. $(8\sqrt{3})^2 = 8^2 * (\sqrt{3})^2 = 64 * 3 = 192$

Substituting these values back into the equation:

$768 = MP^2 + 192$

Now, we need to isolate $MP^2$ by subtracting 192 from both sides:

$MP^2 = 768 - 192$

$MP^2 = 576$

Finally, we take the square root of both sides to find MP:

$MP = \sqrt{576}$

$MP = 24$

Therefore, the height of the equilateral triangle MNO is 24 units.

Understanding the Significance

The height of an equilateral triangle is not just a numerical value; it holds geometric significance. It represents the perpendicular distance from a vertex to the opposite side and plays a crucial role in determining the area of the triangle. The area of a triangle is given by:

$Area = (1/2) * base * height$

In the case of triangle MNO, the base is $16\sqrt{3}$ units, and the height is 24 units. Therefore, the area would be:

$Area = (1/2) * 16\sqrt{3} * 24 = 192\sqrt{3}$ square units.

This connection between the height and the area highlights the importance of accurately calculating the height. Moreover, the height is also related to the inradius and circumradius of the triangle, further emphasizing its significance in geometric calculations.

Alternative Approach: Using 30-60-90 Triangle Properties

There's an alternative approach to solving this problem, which leverages the properties of 30-60-90 triangles. When we draw the altitude in an equilateral triangle, we create two 30-60-90 right triangles. In a 30-60-90 triangle, the sides are in the ratio 1:$\sqrt{3}$:2. The side opposite the 30-degree angle is half the hypotenuse, and the side opposite the 60-degree angle is $\sqrt{3}$ times the side opposite the 30-degree angle.

In our case, the side opposite the 30-degree angle (PO) is $8\sqrt{3}$ units. The side opposite the 60-degree angle (MP) is the height we want to find. Therefore:

$MP = PO * \sqrt{3} = 8\sqrt{3} * \sqrt{3} = 8 * 3 = 24$ units.

This method provides a quicker way to arrive at the answer if you are familiar with the properties of 30-60-90 triangles.

Conclusion: Mastering Geometric Problem-Solving

In conclusion, finding the height of an equilateral triangle involves a blend of geometric understanding and algebraic manipulation. We've explored the solution using the Pythagorean theorem and also touched upon the alternative approach using 30-60-90 triangle properties. The key takeaway is the ability to visualize the problem, break it down into manageable parts, and apply the appropriate theorems and formulas. This exercise not only reinforces our understanding of equilateral triangles and the Pythagorean theorem but also hones our problem-solving skills, which are invaluable in various mathematical and real-world contexts. By mastering these techniques, we can confidently tackle a wide range of geometric challenges and appreciate the beauty and elegance of mathematical reasoning.

Having meticulously calculated the height of the equilateral triangle MNO, which measures 24 units, it's time to align our findings with the provided options and select the correct answer. The options presented are:

A. 12 units B. 24 units C. 36 units D. 72 units

Our calculated height of 24 units perfectly matches option B. Therefore, the correct answer is B. 24 units.

Final Thoughts on Equilateral Triangles and Geometry

Solving this problem has been a comprehensive journey through the world of equilateral triangles and the application of the Pythagorean theorem. We've not only arrived at the correct answer but also deepened our understanding of the properties of equilateral triangles and the significance of the altitude. This exercise serves as a testament to the power of geometric reasoning and the interconnectedness of mathematical concepts. By mastering these fundamental principles, we equip ourselves with the tools to tackle a wide array of geometric challenges and appreciate the elegance and beauty of mathematics.