Question

The legs of a right triangle are in the ratio of 3 to 1. If the length of the hypotenuse of the triangle is 40√40, then the perimeter of the triangle is between
A. 14 and 15
B. 13 and 14
C. 12 and 13
D. 11 and 12
E. 10 and 11

Answer 1

Answer:

$P = 6+2+ \sqrt{40}=14.32$

So then the best option on this case is:

A. 14 and 15

Step-by-step explanation:

When we have a right triangle we can use the Pythegorean identity given by:

$Hip^2 = Opp^2 +Adj^2$  (1)

Where Hip represent the hypothenuse. Opp represent the opposite side and Adj the adjacent side.

On this case we have given the hypothenuse assumed $Hip = \sqrt{40}$ because is the only possible reasonable value because $40\sqrt{40}$ is a too much higher value

We have a ratio provided on this case, let's assume that the ratio is:

$\frac{Opp}{Adj} = 3$

$Opp = 3 Adj$

If we rpelace this condition into equation (1) we got:

$(\sqrt{40})^2 = (3Adj)^2 + Adj^2$

And then we have this:

$40 = 9 Adj^2 + Adj^2 = 10 Adj^2$

$Adj = \sqrt{\frac{40}{10}}=2$

And then the opposite side is:

$Opp = 3 Adj =3* 2= 6$

The perimeter is defined as the sum of all the sides, we can find the perimeter like this:

$P = Opp + Adj + Hip$

And replacing the values that we found we got:

$P = 6+2+ \sqrt{40}=14.32$

So then the best option on this case is:

A. 14 and 15