Two legs of a right triangle are in the ratio 3:4. Find the shorter of the leg if the hypotenuse is 75

Question

I know that the answer is 45 but I can't seem to figure out how I get that answer.

Answer 1

I will list two explanations/methods. If you are just looking for a simple way to solve it, read the first. If you want to know a slightly deeper way to solve it, read the second. :)

 

First explanation: So you want to solve the problem and don't really want to know how it works in depth. That's simple. You just need to realize that it is a 3:4:5 right triangle. Because the sides are proportional in ANY 3:4:5 right triangle, you can set up a simple proportion equation and solve. In this case, we know that the hypotenuse has to be the "5" and the shortest leg has to be the "3". We just do the following:

3               x

_      =       _

 

5               75

 

3/5 = x/75. Cross multiply and solve to get 45. 

 

Second explanation:

 

If it is a right triangle with legs that are 3:4, this is a 3:4:5 right triangle. This is one case of something called a "Pythagorean Triple". In this particular case, the Pythagorean triple is in a ratio of 3:4:5. Any 3:4:5 triangle will be reducible to this ratio. There are other Pythagorean triples, as Harvey said. You can find them by finding numbers that make the following true: a²+b²=c².

 

The shortest leg is the one that is the 3. We know that the hypotenuse must be the 5 because the hypotenuse of a right triangle will always be the longest side . If you do something called the prime factorization (breaking the hypotenuse, which is 75, into prime factors), you will get 3*5*5 is 75. We do this by dividing 75 by different numbers (typically you try low primes, which are 2, 3, 5, and 7) until we get a number that can't be divided evenly any further. 

 

Now that we have our prime factorization, we need to think a little bit more about the Pythagorean Triple. 3:4:5 is the simplest Pythagorean Triple. Any others (of this type with ratio 3:4:5) are just the 3, 4, and 5 multiplied by a coefficient. This number will be the same for all three sides. For example, a right triangle with sides 6, 8, 10 is just the base 3:4:5 triangle with all sides multiplied by 2. 

 

We know that the hypotenuse is the "5" in the 3:4:5. We also know that there is some other number multiplied by the 5 to get 75. Since 3:5:5 is 75's prime factorization and we know that a 5 will be in there because it is a 3:4:5 Pythagorean Triple, we can divide out a single 5 to see what the coefficient has to be. We do that and get 15. This means that the other two sides are 15*3 and 15*4 to get 45 for the shortest side and 60 for the other. 

 

Hope this helps! Sorry for any confusion and thanks again Harvey for pointing out my lack of clarity.