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Geometry/Calculating a Side of Oblique Triangles

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Angles
Angles  
QUESTION: Good Morning, Azeem,

I want to calculate one side of a triangle when I know only the length of one of the other sides and an angle.  Please see the attachment for two situations I am encountering.  One is an obtuse triangle, the other one acute.

Is there a formula?

Thanks.

Pete

ANSWER: Hi Peter,

One side and one angle are insufficient to determine the length of another side.  (Note that there is a range of possible triangles that use that side length and angle measure.)

If you had an angle and two sides, the Law of Cosines could be applied.  Unfortunately, with one angle and only one side, the length of another side cannot be determined.

Thanks for asking,
Azeem

---------- FOLLOW-UP ----------

angles2
angles2  
QUESTION: Hi Azeem:

The way I see it--and tell me if my analysis is correct or not--I'll have to create right triangles.

The triangle on the right is easy:  I draw a horizontal line from the rightmost vertex to the opposite (vertical) side.  Then I have two right triangles and I can apply the tangent function to the two angles to get the length of the vertical side, since I know the length of my horizontal line.

On the left triangle I extend the vertical side down using a dashed line, draw a horizontal line from the rightmost vertex to the dashed line.  Again I use two right triangles (actually the same right triangle twice).  I obtain the total length of the vertical side using tangent, then subtract the length of the dashed line with the tangent function.

This will yield the length of the vertical side of the left triangle, correct?

Thanks.

Pete

ANSWER: Hi Pete,

Nice job with the diagrams.  It really adds clarity to your question.
Given all the information in the diagrams, yes, your analysis is correct.

Understand, however, that this is a different problem than what you had originally asked.  You are assuming you know the length of the newly drawn side and the measures of the two angles it forms.  If this is not clear to you, ask another follow-up and I will go into detail.

Best regards,
Azeem

---------- FOLLOW-UP ----------

QUESTION: Hi Azeem

Yes, I will know the length of the new side via tape measure, and will have use of a clinometer to know the angles.  BTW this problem involves measuring tree heights in a forest, especially when the trees are upslope or downslope.  

One more item to consider:  the right angles will have to be measured correctly and I assume a clinometer has a built-in level for this.  If not, however, I think there can be room for a small amount of error when estimating tree heights.

Answer
Hi Pete,

There might be more error in the use of the tape measure.  In the downslope scenario, you need to attach the tape measure to the tree at a certain height to obtain the horizontal measure.  In the upslope scenario, because of the protracted height, measuring the horizontal might be tough because it is underground.

Just some things to consider.

Good luck!
Azeem

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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