teacher reccomended me

[ryansmith069]

Hi. My name is Ryan Smith. My teacher told me to come here and get help and stuff with my homework. We have 52 kids in my class and she said it is hard for her to always be there for 1 on 1 and this site is the next best thing. I hope someone can help me. Here is the problem I am stuck on.

Find the length, in inches, of the larger side of a right triangle with an area of 210 square inches if its hypotenuse is 29 inches long.


Thanks for the help!

 

[jacket81]

problem

Okay, you can solve this problem using the formula for finding area of a triangle,
and pthy. theorm (a^2+b^2=c^2 where a,b,c are sides of a triangle), and some substitution.
And you'll end up with a equation that at first looks like its impossible to solve, but if you think about it, you can actually use the quadratic formula to solve it!

If you need some more help, ill help you solve it, but i wanted to see what you can come up with!

 

[soroban]

Welcome aboard, Ryabn!

Welcome aboard!

Find the length, in inches, of the larger side of a right triangle with an area of 210 square inches
if its hypotenuse is 29 inches long.

Let the two sides of the right triangle be \(\displaystyle a\) and \(\displaystyle b.\)

The area of the right triangle is: .\(\displaystyle A\:=\:\frac{1}{2}ab\)
Since the area is 210 sq.in., we have: .\(\displaystyle \frac{1}{2}ab\:=\:210\;\;\Rightarrow\;\;b\:=\:\frac{420}{a}\) .[1]

From Pythagorus, we know that: .\(\displaystyle a^2\,+\,b^2\:=\:29^2\) .[2]

Substitute [1] into [2]: .\(\displaystyle a^2\,+\,\left(\frac{420}{a}\right)^2\:=\:841\;\;\Rightarrow\;\;a^2\,+\,\frac{176,400}{a^2}\:=\:841\)

Multiply by \(\displaystyle a^2:\;\;a^4\,+\,176,400\:=\:841a^2\;\;\Rightarrow\;\;a^4\,-\,841a^2\,+\,176,400\:=\:0\)

This factors: .\(\displaystyle (a^2\,-\,400)(a^2\,-\,441)\:=\:0\)

. . and has roots: .\(\displaystyle a^2\,=\,400,\;441\;\;\Rightarrow\;\;a\,=\,\pm20,\,\pm21\)

Since the side of a triangle is a positive number: .\(\displaystyle a\,=\,20,\,21\)

Substituting into [1], we find that: .\(\displaystyle b\,=\,21,\,20\)

In either case, the larger side is 21.