A triangle has sides of 3 cm and 4 cm and one angle of 90° What is the length of the third side?

Question

A triangle has sides of 3 cm and 4 cm and one angle of 90° What is the length of the third side? Without Pythagorean Theorem.

Answer 1

If your triangle looks like this:

Then you can use the trig ratio "tangent" to say:

tan(α) = 3/4 and then solve for m∠α as follows:

tan(α) = 3/4

tan^-1(tan(α)) = tan^-1(3/4)

m∠α = tan^-1(3/4)

Plug into a calculator to get:

m∠α = 36.8699°

Then you can use the trig ratio "sine" as follows:

sin(36.8699°) = 3/h

h•sin(36.8699°) = 3

h = 3/sin(36.8699°)

Plug into a calculator to get:

h = 5

Answer 2

Hi Inara G

Are you referring to finding the Hypotenuse without using the Pythagorean Theorem? If so given the two other legs you can use the Trigonometric Ratios

You don’t necessarily have to use the Pythagorean Theorem, if you don’t want to you should clarify that you don’t want it involved in any way. I mention that because people know how to look up the triples and just give you an answer.

Since you do have a Right Triangle you can use the ratio of the sides

Furthermore if you are asking about the hypotenuse or longest side you find the angles associated with the other legs given

One of the angles will have a tangent of 4/3 while the other will have a tangent of 3/4 if you use your calculator or look up the Tangents to find the angles you should get ~ 53.1 degrees and ~36.9 degrees if you want to know the sin of 36.9 degrees then

sin(36.9°) = 3/Hypotenuse

Hypotenuse = 3/sin(36.9°) = 4.9965 rounding gives you 5.

Another scenario is see if the sides given correspond to any of the known Right Triangles

The info does not correspond to a 30 60 90 this one would have sides corresponding to x, x√3, and 2x; where x is the shortest side

The info does not correspond to a 45, 45, 90 this one would have two equals sides

Also the sum of any two sides of a triangle must be greater than the third side so you know that

3 + 4 = 7 so your third side must be less than 7 . If you look further into this rule and the types of triangles, you will find that a 3, 3, 4 Triangle is an Acute Isosceles Triangle not a Right Triangle. A 4, 4, 3 triangle is an acute Isosceles Triangle again not a Right Triangle.

Again if you are looking for the hypotenuse of your Right Triangle without using the Pythagorean Theorem I’d recommend using the Trigonometric ratios to get the tangents from the other legs, use your calculator to find the angles corresponding to those tangents, then finally use the sine or even the cosine of one of the applicable angle to calculate the hypotenuse.

Hope this helps.