The side of a triangle have lengths of x, x+4, and 20. Specify those values of x which the triangle is acute with longest side 20.
The triangle is acute.
To determine the range of possible values for x such that the triangle is acute and has a longest side of 20, we need to consider the triangle inequality theorem and the conditions for an acute triangle.
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have x, x+4, and 20 as the side lengths. Applying the triangle inequality theorem, we can write:
x + (x + 4) > 20 (1)
x + 20 > x + 4 (2)
x + 4 + 20 > x (3)
Simplifying equation (2) and (3), we get:
2x + 4 > 20 (4)
24 > x (5)
Combining equations (1) and (5), the inequalities we have are:
24 > x > -4
Therefore, the range of values for x that make the triangle acute and with a longest side of 20 is -4 < x < 24.