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Geometry/30-60-90 Triangle

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QUESTION: The length of the hypotenuse of a 30-60-90 triangle is 8. Find the perimeter.

I think it's 12+4 3squared I dunno
Show me how to do this!

ANSWER: Hi C.M.,

The triangle has a right angle, so we can employ basic trigonometry.
Let's say the base angle is 30.  The sine of an angle is the ratio of lengths of sides opposite/hypotenuse, and the cosine of an angle is adjacent/hypotenuse.  Furthermore, sin(30)=1/2 and cos(30)=√2/2.  Use these equations to find the lengths of the adjacent and opposite sides.  Finally, add all three side lengths to get the perimeter.

(You could instead find one of the sides, then use the Pythagorean Theorem to find the other, but the way I described above is faster.)

All the best,
Azeem

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

QUESTION: I still don't understand, could you write out the steps please? Tell me how to use the equations. How can I find the sides of a triangle with the angles of it?

Answer
Hello,

Recall that sin(x), by definition, is opposite over hypotenuse.
sin(x)=opp/hyp

Now substitute in the given information.  Let's focus on the 30-degree angle and the hypotenuse of length 8.
sin(30)=opp/(8)

Rearrange to solve for opposite.
8sin(30)=opp

If possible, simplify.  Here we use that sin(30)=1/2.
8(1/2)=opp
4=opp

Similarly, because cos(x)=adj/hyp, we get that
(hyp)cos(x)=adj
(8)(√2/2)=adj
4√2=adj

Now that we have the lengths of all three sides, we can add them to obtain the perimeter of the triangle.


Thanks for asking,
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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