Comparing Line Segments in Triangles

In summary, the conversation discussed the difference between line segments in triangles, specifically the altitude, median, and bisector. The altitude is a line from a vertex to the opposite side, the median is a line from a vertex to the middle of the opposite side, and the bisector is a line that divides the angle in half. These are all distinct from each other.
  • #1
zeion
466
1

Homework Statement



So these are line segments in triangles. I don't understand how they are different.

Homework Equations





The Attempt at a Solution

 
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  • #2


1zgldu.jpg


The altitude is a line from some vertex to the other side of the triangle (first pic). The median is a line from some vertex to the middle of the side across from it (middle pic). The bisecting angle or whatever it's called is simply a line cast in the direction of 1/2 the angle (last pic)
 
  • #3


Actually, I found a much better image than my mspaint thing.

image003.gif


Consider this triangle ABC

The line from A, straight down is the altitude. The one from A to the middle of the base (close to the a), is the median, and the one that is 1/2 the angle of BAC is the bisector.
 
  • #4


So the median is not the same as the angle bisector?
 

1. What is the Pythagorean Theorem and how is it used to compare line segments in triangles?

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem can be used to compare line segments in triangles by determining whether a triangle is a right triangle and using the lengths of the sides to find the missing length.

2. How do you use the Law of Cosines to compare line segments in triangles?

The Law of Cosines states that in a triangle with sides a, b, and c, the square of the length of side c is equal to the sum of the squares of sides a and b, minus two times the product of the lengths of sides a and b multiplied by the cosine of the angle opposite side c. This law can be used to compare line segments in triangles by finding the missing length or angle in a non-right triangle.

3. Can you compare line segments in any type of triangle?

Yes, line segments can be compared in any type of triangle. However, the methods used may differ depending on the type of triangle. For example, the Pythagorean Theorem can only be used to compare line segments in right triangles, while the Law of Cosines can be used for any type of triangle.

4. How can you tell if two line segments in a triangle are equal in length?

If two line segments in a triangle are equal in length, then the triangle is either an isosceles triangle or an equilateral triangle. This means that two sides of the triangle are equal in length, or all three sides are equal in length, respectively.

5. Are there any other methods besides the Pythagorean Theorem and Law of Cosines to compare line segments in triangles?

Yes, there are other methods to compare line segments in triangles such as the Law of Sines, which relates the lengths of the sides of a triangle to the sine of its angles. Additionally, congruence postulates and theorems can be used to compare line segments in congruent triangles.

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