The Sides Of An Equilateral Triangle Are 8 Units Long. What Is The Length Of The Altitude Of The Triangle?A. $5 \sqrt{2}$ Units B. $4 \sqrt{3}$ Units C. $ 10 2 10 \sqrt{2} 10 2 ​ [/tex] Units D. $16 \sqrt{5}$ Units

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Introduction

An equilateral triangle is a triangle with all sides of equal length. In this problem, we are given that the sides of an equilateral triangle are 8 units long. We are asked to find the length of the altitude of the triangle. The altitude of a triangle is a line segment that connects a vertex of the triangle to the opposite side, forming a right angle. In this case, we are looking for the length of the altitude from one of the vertices to the opposite side.

Understanding the Properties of an Equilateral Triangle

An equilateral triangle has several important properties that we can use to solve this problem. One of the key properties is that all three sides of an equilateral triangle are equal in length. In this case, we are given that the sides are 8 units long. Another important property is that the angles of an equilateral triangle are all equal. Since the sum of the angles in a triangle is always 180 degrees, each angle in an equilateral triangle must be 60 degrees.

Drawing the Altitude

To find the length of the altitude, we can draw a diagram of the equilateral triangle and the altitude. Let's call the vertex from which we are drawing the altitude point A, and the opposite side point B. We can draw a line from point A to point B, forming a right angle. This line is the altitude of the triangle.

Using the Pythagorean Theorem

To find the length of the altitude, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled 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. In this case, the hypotenuse is the side of the equilateral triangle, which is 8 units long. The other two sides are the altitude and the half of the base of the triangle.

Calculating the Length of the Altitude

Let's call the length of the altitude h. We can use the Pythagorean theorem to set up an equation:

h^2 + (8/2)^2 = 8^2

Simplifying the equation, we get:

h^2 + 16 = 64

Subtracting 16 from both sides, we get:

h^2 = 48

Taking the square root of both sides, we get:

h = √48

Simplifying the square root, we get:

h = 4√3

Conclusion

In this problem, we were given that the sides of an equilateral triangle are 8 units long. We were asked to find the length of the altitude of the triangle. Using the properties of an equilateral triangle and the Pythagorean theorem, we were able to calculate the length of the altitude. The length of the altitude is 4√3 units.

Answer

The correct answer is B. $4 \sqrt{3}$ units.

Discussion

This problem is a great example of how to use the properties of an equilateral triangle and the Pythagorean theorem to solve a problem. It also shows how to simplify square roots and calculate the length of a line segment. If you have any questions or need further clarification, please don't hesitate to ask.

Related Topics

  • Equilateral triangles
  • Pythagorean theorem
  • Right-angled triangles
  • Line segments
  • Geometry

References

  • [1] "Geometry" by Michael Artin
  • [2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  • [3] "The Art of Problem Solving" by Richard Rusczyk

Additional Resources

  • Khan Academy: Equilateral triangles
  • Khan Academy: Pythagorean theorem
  • Mathway: Equilateral triangle altitude
  • Wolfram Alpha: Equilateral triangle altitude
    Frequently Asked Questions: Equilateral Triangles and Altitudes ====================================================================

Q: What is an equilateral triangle?

A: An equilateral triangle is a triangle with all sides of equal length. In other words, all three sides of an equilateral triangle are equal in length.

Q: What is the altitude of an equilateral triangle?

A: The altitude of an equilateral triangle is a line segment that connects a vertex of the triangle to the opposite side, forming a right angle.

Q: How do I find the length of the altitude of an equilateral triangle?

A: To find the length of the altitude of an equilateral triangle, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled 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.

Q: What is the formula for finding the length of the altitude of an equilateral triangle?

A: The formula for finding the length of the altitude of an equilateral triangle is:

h^2 + (s/2)^2 = s^2

where h is the length of the altitude, s is the length of the side of the equilateral triangle, and s/2 is the length of the half of the base of the triangle.

Q: How do I simplify the square root of a number?

A: To simplify the square root of a number, you can look for perfect squares that divide the number. For example, if you have √48, you can simplify it by looking for perfect squares that divide 48. In this case, you can simplify √48 as 4√3.

Q: What is the length of the altitude of an equilateral triangle with sides of length 8 units?

A: To find the length of the altitude of an equilateral triangle with sides of length 8 units, you can use the formula:

h^2 + (8/2)^2 = 8^2

Simplifying the equation, you get:

h^2 + 16 = 64

Subtracting 16 from both sides, you get:

h^2 = 48

Taking the square root of both sides, you get:

h = √48

Simplifying the square root, you get:

h = 4√3

Q: What are some real-world applications of equilateral triangles and altitudes?

A: Equilateral triangles and altitudes have many real-world applications, including:

  • Architecture: Equilateral triangles are often used in the design of buildings and bridges.
  • Engineering: Equilateral triangles are used in the design of machines and mechanisms.
  • Art: Equilateral triangles are used in the design of patterns and shapes.
  • Science: Equilateral triangles are used in the study of geometry and trigonometry.

Q: How can I practice solving problems involving equilateral triangles and altitudes?

A: You can practice solving problems involving equilateral triangles and altitudes by:

  • Using online resources, such as Khan Academy and Mathway.
  • Working with a tutor or teacher.
  • Practicing with sample problems and exercises.
  • Joining a study group or math club.

Conclusion

In this article, we have discussed the properties of equilateral triangles and altitudes, and provided answers to frequently asked questions. We have also provided a formula for finding the length of the altitude of an equilateral triangle, and explained how to simplify square roots. We hope that this article has been helpful in understanding the concepts of equilateral triangles and altitudes. If you have any further questions or need additional clarification, please don't hesitate to ask.