SOLUTION: The length of the longest side is the integer given. What value(s) of x make the triangle? x-4 , x+7, 45 ; Acute triangle

Algebra ->  Triangles -> SOLUTION: The length of the longest side is the integer given. What value(s) of x make the triangle? x-4 , x+7, 45 ; Acute triangle      Log On


   

Question 1191818: The length of the longest side is the integer given. What value(s) of x make the triangle? x-4 , x+7, 45 ; Acute triangle
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Finding the maximum possible value of x is easy. It is given that the longest side is 45, so

x%2B7%3C45
x%3C38

To determine the smallest possible values of x, note that the triangle is a right triangle if

%28x-4%29%5E2%2B%28x%2B7%29%5E2=45%5E2

So the triangle is acute if

%28x-4%29%5E2%2B%28x%2B7%29%5E2%3E45%5E2

%28x-4%29%5E2%2B%28x%2B7%29%5E2=45%5E2
x%5E2-8x%2B16%2Bx%5E2%2B14x%2B49=2025
2x%5E2%2B6x%2B65=2025
2x%5E2%2B6x-1960=0
x%5E2%2B3x-980=0

Use the quadratic formula to find

x=%28-3%2Bsqrt%283929%29%29%2F2 = 29.805 to 3 decimal places

So the triangle with longest side 45 and the other two sides x-4 and x+7 is acute for any value greater than (approximately) 29.805 and less than 38.

ANSWER: 29.805 < x < 38