I need help with a homework question. The perimeter of a right triangle is 24 meters. If the lengths of the sides are two consecutive even integers, what is the area of the right triangle? Thanks for anyone's help.
Let a,b,c be the lengths of the sides.
By the Pythagore theorem, you can suppose c the longest side and c²=a²+b² (1), and you know that a+b+c=24 (2)
The area of the triangle is ab/2 (because you've supposed a and b the sides of the right angle)
a and b are two consecutive even numbers. So you can write a is 2n and b is 2n+2.
ab/2 will be (2n*(2n+2))/2 = 2n*(n+1)=2n²+2n
By (2), we have 2n+2n+2+c=24 => c=22-4n => c²=484+16n²-176n
By (1), we have c²=4n²+4n²+4+8n=8n²+8n+4
=> n=20 (impossible because a+b+c=24) or n=3, and you replace in the area formula
(=> a=6 and b=8)
Thank you for your help with the right triangle question. I have another if you do not mind.
A 120 meter board is cut into 3 pieces. The second piece is 40 meters longer than the first piece and the third piece is two times as long as the second piece. Find the length of the longest piece.
I don't know whether the answer should be "no solution" because nothing can be 0m in length or "80m" to simply answer the question and ignore the fact that there is no first piece. Thank you very much!