Geometry: Triangle areas with apothems.

little.red

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I have a triangle that I need to solve for its area and I am given a side length of 15.6 and an apothem of 4.5. I'm completely lost.
 
I have a triangle that I need to solve for its area and I am given a side length of 15.6 and an apothem of 4.5. I'm completely lost.
More information is needed.

The "apothem", in this context, is the line from the "center" of the triangle, down to the midpoint of the base (the horizontal line at the bottom) of the triangle. This line may or may not be the height.

The triangle has three sides. The given side may be any of those three.

Unless the side happens also to be the base and the apothem happens also to be along the (that is, containing the) height's line, the area cannot be determined.

Please reply with the rest of the information or, if there is none, please first consult with your instructor.

When you reply, please include a clear listing of your efforts so far. Thank you! ;)
 
Last edited:
Thanks to Ishuda for correcting me via private message. :oops: In particular:

The "apothem" (or "in-radius") goes from the center of a regular figure to the midpoint of one of the equal-length sides. As such, the given side-length is the length of all three sides, and the triangle must be equi-lateral / equi-angular.

If you haven't been given a formula for working with side-lengths and apothems, you can figure things out for yourself:

Draw the equilateral triangle, with its base at the bottom being an horizontal line. Draw a dot somewhere in the middle for the center; name this point "C". Draw a vertical line from this dot down to the midpoint of the base; label the midpoint as "M". Draw a slanty line from the center to the left-hand end of the base. (The left-hand end isn't special; I'm just wanting us to be looking at the same picture.) Label this left-hand end "L".

By nature of the apothem, the line segment CM is perpendicular to the line segment LM. So CML is a right triangle. What is the length of the base of this triangle? What is the height of this triangle? What then is the area of this triangle?

If you looked at the other five triangles that you can form, in this same way, inside the original triangle, how would their areas compare? What then is the area of the entire original triangle? ;)
 
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