Stuipd Simple Geometry Question

[dmt740]

Homework Statement

Ok, I need to find the dimensions of a triangle. I have the length of the hypotenuse(sp?) and a ratio of the other sides. And yes, in case you are wondering, I am figuring out TV dimensions. :-/

Homework Equations

If using a+b+c, I know c, and the values of A and B are a 16:9 ratio, what's the equation to solve?

The Attempt at a Solution

Dunno, that's why I am here. Forgive the simple question, but if you don't ask you won't know. TIA.

 

[Mentallic]

I'm assuming you're looking for the values of a and b.

If you have the the lengths a:b in the ratio 16:9 then you have:

$$\frac{a}{b}=\frac{16}{9}$$

and you're also using a right-angled triangle, so use pythagoras' theorem:

$$a^2+b^2=c^2$$

So all you need to do is solve these two equations simultaneously since there are two variables to find (c is a constant since you know what it is).

 

[dmt740]

Forgive me but my algebra days are long gone. I remember some things but I couldn't figure out how to solve those two equations together. I just kept going round in circles.
So there has to be an easier way than what I figured out.
9^2 + 16^2 = c^2
81 + 256 = 337
then I took those number and made my own ratio
337 / 256 = 1.31640625
337 / 81 = 4.16049382
take those answers and use a different value for c, let's say 32
32 * 32 = 1024
1024 / 1.31640625 = 777.8753709
777.8753709^1/2(sqrt) = 27.890417 which equals b

1024 / 4.16049382 = 246.1246290
246.1246290^1/2(sqrt) = 15.6883596 which equals a

Then I just played around and found that a 15 by 28 measurement gives 31.7 which is actually what a 32" TV is measured at.

Again I say, there has to be an easier way.

 

[Bohrok]

I couldn't quite follow everything you did, and there is a more straightforward way, so:

The ratio of the length and width of the TV is 16:9 , or let a/b = 16/9. We don't know what a or b are, so let's not assume or work with them as 16 or 9.
$$\frac{a}{b} = \frac{16}{9} \Longrightarrow a = \frac{16b}{9}$$
Using the Pythagorean theorem for a TV with length and width a and b, and a diagonal length c of 32, we have a² + b² = 32² = 1,024.

Substitute a = 16b/9 into the previous equation to get
$$\left(\frac{16b}{9}\right)^2 + b^2 = 1024$$

Solve for b, then substitute that value into a = 16b/9 to get a, and you'll have the width and length of a 32" TV screen.

 

[Mentallic]

I am quite astounded at your approach to the problem. Especially since it simply worked
Such methods were probably the logic used before algebraic equations had been "invented".

 

[dmt740]

I couldn't quite follow everything you did, and there is a more straightforward way, so:

The ratio of the length and width of the TV is 16:9 , or let a/b = 16/9. We don't know what a or b are, so let's not assume or work with them as 16 or 9.
$$\frac{a}{b} = \frac{16}{9} \Longrightarrow a = \frac{16b}{9}$$
Using the Pythagorean theorem for a TV with length and width a and b, and a diagonal length c of 32, we have a² + b² = 32² = 1,024.

Substitute a = 16b/9 into the previous equation to get
$$\left(\frac{16b}{9}\right)^2 + b^2 = 1024$$

Solve for b, then substitute that value into a = 16b/9 to get a, and you'll have the width and length of a 32" TV screen.

I was trying to do that, but I couldn't remeber how to square a fraction with a variable in it. I think I did figure it out, finally.

I am quite astounded at your approach to the problem. Especially since it simply worked
Such methods were probably the logic used before algebraic equations had been "invented".

Welcome to most of my mathematics career. I can get the right answer, just not the conventional way. That caused a few issues with certain teachers.