For how many integers x does a triangle with side lengths 10,29 and x have all its angles acute. how do I know which side is the hypotenuse or does it not matter? Please give me an explanation.
the law of cosines says that if x is the longest side and θ is the opposite angle,
x^2 = 10^2 + 29^2 - 2*10*29 cosθ
If x is less than 10^2+29^2, all three angles are acute.
Of course, if x is not the longest side, then rearrange the numbers a bit to find other conditions.
To determine the number of integers, x, such that a triangle with side lengths 10, 29, and x has all acute angles, we need to examine the possible values of x.
In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem. Mathematically, we can express this as:
x + 10 > 29
x + 29 > 10
10 + 29 > x
Let's solve these inequalities one by one:
1. x + 10 > 29
Subtract 10 from both sides:
x > 19
2. x + 29 > 10
Subtract 29 from both sides:
x > -19
3. 10 + 29 > x
Combine the terms:
39 > x
From the above inequalities, we can conclude that x must be greater than 19 and less than 39.
Now, let's consider your question about which side is the hypotenuse. In a triangle, the hypotenuse is the longest side, and it is opposite the largest angle. However, since you mentioned acute angles, we can conclude that the triangle is not a right triangle, which means it does not have a hypotenuse.
In this case, we are simply dealing with a triangle with three sides of lengths 10, 29, and x, where x is the value we are interested in to ensure acute angles. As we established earlier, x must be greater than 19 and less than 39.
To determine the number of integers within this range, we can simply count the number of integers between 19 and 39 (inclusive).
Therefore, there are 21 integers (including both 19 and 39) for which a triangle with side lengths 10, 29, and x has all acute angles.