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.
To determine the number of integers x for a triangle with side lengths 10.29 and x to have all acute angles, we need to consider the triangle inequality theorem. According to this theorem, in a triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the remaining side.
In this case, we have side lengths 10.29 and x. For a triangle to have all acute angles, we need x to be less than the sum of 10.29 and 10.29 (which is 20.58). So, we can write the inequality:
x < 20.58
To count the number of integers x that satisfy this inequality, we need to find the set of integers satisfying the condition. In this case, we have an upper limit for x (20.58), so we need to find the number of integers less than this upper limit.
To count the number of integers less than a given number, we subtract 1 from the upper limit and floor the result (i.e., round down to the nearest whole number since x must be an integer in this case). So, we get:
Number of integers x < 20.58 = floor(20.58) - 1 = 20 - 1 = 19
Thus, there are 19 integers for x that satisfy the conditions for a triangle with side lengths 10.29 and x to have all acute angles.
Now, regarding your question about which side is the hypotenuse, it is important to note that the term hypotenuse refers specifically to the side opposite the right angle in a right triangle. In this case, since the question doesn't mention any right angles, we are not dealing with a right triangle and there is no hypotenuse. The terminology regarding sides in triangles is generally used for right triangles. For non-right triangles, we usually refer to them as sides or edges rather than specifically using terms like hypotenuse.