two sides of a triangle measure 8 and 15. how many integer values can be assigned to the third side so that all the angles in the triangle are acute.
For all angles to be acute, no single angle can be ≥ 90°
let's assume the angle between the 8 and 15 is 90°
by Pythagoras, the third side would have to be17.
So the largest integer we could have for the third side is 16
But to have a triangle , the sum of any two sides has to be greater than the third side.
let the third side be x
then
x+8 > 15 or x > 7
so possible third sides are
8,9,10,11,12,13,14,15,16 or 9 integer values
To determine the values that can be assigned to the third side of a triangle so that all angles are acute, we need to consider the triangle inequality theorem. According to this theorem, 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 two sides measuring 8 and 15. Let's call the third side "x". For the triangle to have all acute angles, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we have two conditions to satisfy:
1) 8 + x > 15
2) 15 + x > 8
Simplifying these inequalities, we get:
1) x > 7
2) x > -7
Since the length of a side cannot be negative, we can ignore the second inequality. Thus, the only condition we need to satisfy is x > 7.
Now, let's consider the range of integer values that can be assigned to x. Since we want the length of the third side to be an integer value, we need to take the ceiling of the lower bound of our condition.
Ceiling(7) = 8
Therefore, we can assign any integer value greater than or equal to 8 to the third side so that all the angles in the triangle are acute.