Triangle MNO is an equilateral triangle with sides measuring $16 \sqrt{3}$ units. What is the height of the triangle?
A. 12 units
B. 24 units
C. 36 units
D. 72 units
Answer (1)
Recognize that the height of an equilateral triangle bisects the base, forming two 30-60-90 right triangles.
Apply the formula h = 2 3 × s to calculate the height, where s is the side length.
Substitute the given side length s = 16 3 into the formula.
Calculate the height: 24 units.
Explanation
Problem Analysis We are given an equilateral triangle MNO with side length 16 3 units. Our goal is to find the height of this triangle.
Height Formula In an equilateral triangle, the height bisects the base, creating two 30-60-90 right triangles. The height is opposite the 60-degree angle. We can use the formula for the height of an equilateral triangle, which is:
h = 2 3 × s
where h is the height and s is the side length of the equilateral triangle.
Calculate the Height Substitute the given side length s = 16 3 into the formula:
h = 2 3 × ( 16 3 )
h = 2 3 × 16 3
h = 2 16 × 3
h = 2 48
h = 24
Therefore, the height of the equilateral triangle is 24 units.
Final Answer The height of the equilateral triangle MNO is 24 units.
Examples
Equilateral triangles are not just theoretical concepts; they appear in many real-world applications. For example, architects use equilateral triangles in designing structures for stability and even aesthetics. Imagine designing a bridge where equilateral triangles form the basic support structure. Knowing how to calculate the height of such triangles allows engineers to determine the necessary dimensions and ensure the bridge's structural integrity. In this case, if the sides of the triangular supports are 16 3 meters, the height would be 24 meters, which is crucial for calculating load distribution and overall stability.
