Can sides A + B of a triangle be found if side C is known and...?

Question

Can sides A + B of a right triangle be found if side C is known and you know that the ratio of the other two sides is 16 : 9 ?

I know that A^2 + B^2 = C^2, and normally you'd need to either know 2 of the lengths of the sides and/or an angle other than the right angle.

This is the question I'm trying to answer:
If I know that a high-defininition (widescreen) television has a 61" (diagonal) screen, and I also know that the aspect ratio for this type of television is 16 (horizonal) : 9 (vertical), can I solve for the other sides? I've tried using the 3-4-5 model to solve this and it does not apply. Can the actual width of the television be calculated in inches given this information?

This would be a good problem for a math test - feel free to use it, teachers!

Answer 1

Yes, of course.

(16x)² + (9x)² = 61²
256x² + 81x² = 3721
337x² = 3721
x² = 11.04
x = 3.323"

so the longer side is 16(3.323) = 53.166" and the shorter is 9(3.323) = 29.906".

Answer 2

First, I'd like to mention that the "3-4-5 model" produces a screen with about 10% more area and this is why all laptops now come with a widescreen display. It is not because the manufacturers think everyone wants to use them as expensive portable DVD players, but rather that they have foisted a 10% smaller piece of screen on us while convincing us it is better and therefore worth more of our cash! Trouble is, most of us still work with "up and down" things rather than wide things so we either see less of our work at a time or have to see it smaller to fit in the now shorter height! (Sigh...)

That said...

of course you can figure this out! And not by using the obvious path of trig functions that require a calculator or tables. If you remember that similar triangles have sides in proportion and you know one measurement in the triangle of choice, your TV. So set up a second triangle with measurements of 16 and 9 on the sides adjacent to the right angle and call those width and height. So now 16^2 + 9^2 = diagonal measurement squared = 256 + 81 = 337. So the second triangle's hypotenuse (diagonal) meausres the square root of 337 ( 337^½).

Then use the fact of proportionality to figure your TV's screen width and screen height:

Width: 337^½:61 :: 16:width so: width = 16 * 61 / 337^½ = 53 inches (to 2 significant digits)

Height: 337^½:61 :: 9:height so: height = 9 * 61 / 337^½ = 30 inches (to 2 significant digits)

So it is 30" by 53".

Answer 3

this is not a problem. since you have 16:9, the third side is:
(16^2 + 9^2)^(1/2) = X

your diagonal is 61. take the ratio of your diagonal to X (X/62) and multiply it by 16 and 9 to get the length of the other sides.

corresponding parts of similar triangles are similar

Answer 4

if it is a right triangle, you can use a^2 + b^2 = c^2
otherwise you have to know at least one of the angles of the triangle to use cosine of sine law to find the third edge


your example is an right angle so you can use Pythagorean Theorem

we have :
9a = 16b and a^2 + b^2 = c^2 = 61^2
<=> (256/81 + 1)* b^2 = 61^2 <=> b = 29.906ft
<=> a = 53.166ft

Answer 5

You are already on the right track. You know that the hypotenuse of the right triangle is 61", which is C^2. Plug in the other figures: A^2 + B^2 = 61.

Answer 6

Actually it is kind of easy.

Set the short side to x:
the long side would be x*16/9:
Your formunla would be x^2 + (x*16/9)^2 = 61^2.

Solve for X. . .

Answer 7

If you measure at least one angle and calculate the other by the formula a + b + c = 180, then multply 61 by the sin and cos of angles a and c you should get the height & width, but for the like of me I can't remember which angle gets the sin and cos and I left my ti86 at home. My trigonometry is so rusty I have forgotten a lot of it, I do remember that the length and width are related to the sin and cos of the angles.

Answer 8

does not make plenty experience... considering the fact that in a trianble A^2 + B^2 = C^2 and u have been given 12^2 = 8^2 + 5^2 = > one hundred forty four =sixty 4 + 25 so what u have been given is one hundred forty four = 89 that doesn't make plenty experience. In an popular suitable triangle, use Tangent to locate the angles...

Answer 9

Let x be a common factor of both the sides
then

(16x)^2+(9x)^2= 61^2

which gives x=3.32

thus length would be 16*3.32= 53.12
height= 9*3.32= 29.88
just round them up if u want to

Answer 10

i think it would be possible to do that. since you know the ratio of the two sides is 16:9, you can apply the pythagorean theorem:

(16x)^2 + (9x)^2 = c^2 (where you know the value of c

solve for the value of x.. and then you can find the two sides A and B