without trig

[tomas]

[Cap'n Refsmmat]

Is this a homework question or just a brainteaser?

 

All you have to do is remember that the interior angles in a triangle add up to 180 degrees.

[alan2here]

How totally careless of me, ignore this post

 

3x + 4x + 5x = 180

find x

12x = 180

x = 180 / 12

x = ?

Edited June 6, 2008 by alan2here

[Bignose]

3x + 4x + 5x = 180

 

Ummmmm, no. 3x + 4x + (5x + something) = 180.

 

This is along the right idea -- using the sum of the internal angles must be equal to 180 degrees -- but 5x isn't the entire angle of the top of the large triangle.

 

Using the idea that the angles of a triangle must add to 180 you can write a bunch of equations. Also, the angles along a straight line (not so subtle hint, that'd be angle ABD and angle BDC) also add to 180.

 

Write out all the equations for the triangle and the straight line and you should have more than enough equations to determine any unknowns.

Edited June 5, 2008 by Bignose

[ydoaPs]

You can use the fact that the line makes two triangles to help you.

[alan2here]

3x + 4x + 5x + n = 180

12x + n = 180

(x + n = 180 / 12?)

 

3x + 5x + n = 180

8x + n = 180

 

4x + n + o = 180

 

I'm a little stuck here as well

[ydoaPs]

3x + 4x + 5x + n = 180

12x + n = 180

(x + n = 180 / 12?)

You made a math error here. Also, don't the angles add up to 360 degrees?

 

3x + 5x + n = 180

8x + n = 180

 

4x + n + o = 180

 

I'm a little stuck here as well

 

Remember that since the line makes two triangles out of one, you now have three triangles. Since there are two unknowns, I made a system of three equations with each equation relating the angles of a triangle. I then solved the system of equations for x using the substitution method.

[Cap'n Refsmmat]

You made a math error here. Also, don't the angles add up to 360 degrees?

The angles of a triangle add up to 180 degrees.

[ydoaPs]

The angles of a triangle add up to 180 degrees.

 

That depends on the geometry you're using!

 

Either way, my method still works.

[Bignose]

4 equations 4 unknowns:

 

right most small triangle:

5x + 3x + BDC = 180

 

left most small triangle:

4x + ABD + ADB = 180

 

large triangle:

4x + ABD + 5x + 3x = 180

 

line:

ADB + BDC = 180

 

Now it's just equation solving...

 

-------------------------

 

I have a small side question based on the original question of the OP.

 

Would knowing that a triangle has 180 degrees be something known "without trig" I mean, trig obviously knows that 180 degrees is special -- sin(180 degrees) = 0, cos(180 degrees) = -1, etc. But, is that 180 degrees something known without trig? To a certain extent, didn't trig help define exactly what a degree, and what a radian is? I don't know the history of math well enough.... these are just ponderances of mine.

[tomas]

but i cant do that

[Bignose]

but i cant do that

 

what exactly can't you do?

[tomas]

4 equations 4 unknowns:

[Bignose]

Here's how you do 2 equations with 2 unknowns:

 

As an example, let's look at:

x+y = 7.2

3x-2y = 10.6

 

Rearrange one of the equations to isolate a variable. I'm going to do the first equation:

y = 7.2-x

 

Now, plug this definition of y into the second equation:

 

3x -2(7.2-x) = 10.6

 

And solve for x

 

Then, with the solution for x, you can compute y = 7.2-x

 

You should do this and find that x=5 and y =2.2

 

Note that the choices are completely arbitrary.

 

I could have used x = 7.2-y as the rearrangement.

 

Or, I could have used the second equation.

 

x = (1/3)*(2y + 10.6)

or

y = (1/2)*(-10.6+3x)

 

Now, 4 equations with 4 unknowns is exactly the same. You just have to repeat it a lot more times. And be careful to write down every step and be accurate to make sure the results are correct.

[Anubisboy]

Bignose, I may be making a mistake, but after solving the equations you provided, I didn't get obtain a result for [math]x[/math].

 

Equation one implies

 

[math] BDC=180-8x[/math],

 

and equation three implies

 

[math]ABD=180-12x[/math].

 

Substituting the result from equation 3 into equation 2 gives

 

[math]4x+ADB+180-12x=180 \Longrightarrow ADB=8x[/math].

 

Substituting the result for [math]ABD[/math] plus the result obtained for [math]BDC[/math] into equation four yields

 

[math] 180-8x+8x=180 \Longrightarrow 180=180[/math], which, while true, is not very revealing.

 

 

 

I think one must make use of the fact that [math]AB=CD[/math]. How this is done is not readily apparent to me.

 

Anubisboy

[Bignose]

Anubis boy, you are right. The 4 equations aren't linearly independent... I too get only trivial results from it -- angles equal to 0 and 180.

 

I don't have a tremendous amount of time to think on this right now, but I'm going to keep mulling it over...

[Anubisboy]

Following the link located at the lower left hand corner of the diagram, I found the problem at this site as well. ([url=http://www.gogeometry.com/problem/problem013.htm][/url]http://www.gogeometry.com/problem/problem013.htm). However, the description of the problem doesn't completely match the diagram for the problem. There must be an error in either the diagram or description.

[DeanK2]

[math]BDC=180-8x[/math]

[math]ABD=180-12x[/math]

 

[math]0<x< 15[/math]

 

[math]20x+ABD+BDC=360; =>...ABD+BDC > 60...if...0<x<15[/math]

 

 

With the limits above, x can be found.

A simple logarithm would suffice.