The length of an altitude of an equilateral triangle with a side length of 6 is 3√3.
Here's how to find it:
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Understanding the Altitude: An altitude of a triangle is a perpendicular line segment drawn from a vertex to the opposite side. In an equilateral triangle, the altitude bisects the base and forms two congruent 30-60-90 right triangles.
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30-60-90 Triangle Properties: In a 30-60-90 triangle, the sides are in the ratio of 1:√3:2. The shorter leg is opposite the 30-degree angle, the longer leg is opposite the 60-degree angle, and the hypotenuse is opposite the 90-degree angle.
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Applying the Properties: In our equilateral triangle, the base is 6, so each half of the base is 3. This is the shorter leg of the 30-60-90 triangle. The altitude is the longer leg, which is √3 times the shorter leg. Therefore, the altitude is 3√3.
Example:
Imagine an equilateral triangle with sides of length 6. Draw an altitude from one vertex to the opposite side. This altitude divides the triangle into two congruent 30-60-90 triangles. The shorter leg of each of these triangles is 3 (half of the base of the equilateral triangle). The altitude is the longer leg, which is 3√3.
Practical Insights:
- The altitude of an equilateral triangle is also its median and angle bisector.
- This formula can be used to calculate the area of an equilateral triangle: (1/2) base altitude = (1/2) 6 3√3 = 9√3.
