Triangle with median and altitude

In summary, the problem involves finding the value of sin^3(A/3)cos(A/3) in a triangle ABC with A as an obtuse angle, where AD and AE are the median and altitude, respectively. Using the three right-angled triangles formed, we can simplify the expression to cos(A/3) = (q^2 + c^2 - (a/2 - x)^2)/2cq, where q and c are known values. However, the exact value of sin(A/3) remains unknown.
  • #1
utkarshakash
Gold Member
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Homework Statement


In triangle ABC with A as obtuse angle, AD and AE are median and altitude respectively. If BAD = DAE=EAC, then sin^3(A/3)cos(A/3) equals

Homework Equations


The Attempt at a Solution



CE = a/2. Let DE = x. Then BD = a/2 - x.
Let AE = p, AD = q.

For ΔADB

[itex]\cos \frac{A}{3} = \dfrac{q^2+c^2-(a/2 -x)^2}{2cq}[/itex]

I can also write cos A/3 for other two triangles using the same approach but I don't know which one to use in the final expression. Also it will contain p and q which is unknown. I have no idea what sin(A/3) would be. This seems really complicated. :cry:
 
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  • #2
hi utkarshakash! :smile:
utkarshakash said:
For ΔADB

[itex]\cos \frac{A}{3} = \dfrac{q^2+c^2-(a/2 -x)^2}{2cq}[/itex]

why use such an awkward triangle? :redface:

there are three right-angled triangles …

use them! :wink:​
 
  • #3
tiny-tim said:
hi utkarshakash! :smile:


why use such an awkward triangle? :redface:

there are three right-angled triangles …

use them! :wink:​

How could I miss them?:tongue2: Thanks a lot.
 

1. What are the properties of a triangle with median and altitude?

A triangle with median and altitude has the following properties:

  • It has three sides, three angles, and three vertices.
  • The median divides each side into two equal segments.
  • The altitude is perpendicular to the base and passes through the opposite vertex.
  • The median and altitude intersect at a point called the centroid, which is two-thirds of the distance from each vertex to the midpoint of the opposite side.
  • The sum of the lengths of any two sides is always greater than the length of the third side, a property known as the triangle inequality.

2. How do you find the length of the median and altitude in a triangle?

To find the length of the median, you can use the formula: m = √(2b^2 + 2c^2 - a^2)/2, where a, b, and c are the lengths of the sides of the triangle. To find the length of the altitude, you can use the formula: h = (2/A) * √(s(s-a)(s-b)(s-c)), where A is the area of the triangle and s is the semi-perimeter (a + b + c)/2.

3. What is the relationship between the median and altitude in a triangle?

The median and altitude in a triangle are perpendicular to each other and intersect at the centroid. The centroid divides the median in a 2:1 ratio, with the longer segment being two times the length of the shorter segment. The altitude from the vertex to the base is also divided in a 2:1 ratio, with the longer segment being two times the length of the shorter segment.

4. How does a triangle with median and altitude differ from a regular triangle?

A triangle with median and altitude differs from a regular triangle in that it has an additional line segment, the median, and an additional perpendicular line segment, the altitude. These segments create the centroid, which is a point of concurrency in the triangle. A regular triangle does not have a centroid or a perpendicular line segment from a vertex to the opposite side.

5. What is the importance of understanding triangles with median and altitude?

Understanding triangles with median and altitude is important in geometry and other fields of science and math. It helps in the calculation of geometric properties and measurements, such as area, perimeter, and angles. It also helps in solving more complex problems involving triangles and can be applied in fields such as engineering, architecture, and navigation. Additionally, understanding triangles with median and altitude can help in visualizing and understanding other geometric concepts, such as the Pythagorean theorem and the concept of similarity.

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