Give the possible values of x, if the following expressions represent the lengths of the sides of a triangle.

Question

a. 3x - 1; 2x + 4; 2x + 3

b. 5x - 5; 3x + 6; 4x + 2

c. 4x - 4; 2x + 6; 3x + 5

Answer 1

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. This rule must be satisfied for all 3 combinations of the sides. So if a, b, and c are the three sides of a triangle, then we must have:

a + b > c

a + c > b

b + c > a

Let's try the first problem, where a = 3x-1, b = 2x+4, and c = 2x+3:

a + b > c

3x-1 + 2x+4 > 2x + 3

5x + 3 > 2x + 3

x > 0

a + c > b

3x-1 + 2x+3 > 2x + 4

5x + 2 > 2x + 4

3x > 2

x > 2/3

b + c > a

2x+4 + 2x + 3 > 3x - 1

4x + 7 > 3x - 1

x > -8

Only values of x > 2/3 satisfy all three x inequalities, so that's the answer.

You try the rest.

Answer 2

Not enough imfo. The formula for 3 sides is: α=acos((b^2+c^2-a^2)/(2bc).

β and γ are figured the same way by using the relative sides.