Please imagine a triangle here. >.< the 'o' is a person and the two peaks are the mountains.
/\___/\
_\___/
___o
Explain why the distance between these two peaks are greater than the difference of the distances from the person to each peak.
In any non-degenerate triangle, the sum of any two sides is always greater than the third side.
Let A,B represent the two peaks, and C the person.
From the first statement, we can conclude that AB+BC>AC, or equivalently AB>AC-BC, which translates to the distance between the two mountains is greater than the difference between the distances from the person to the mountains.
Thanks MathMate.
To explain why the distance between the two peaks is greater than the difference of the distances from the person to each peak, we need to consider the concept of triangle inequality.
In this case, we have a triangle with one side between the two peaks (let's call it AB) and two other sides representing the distances from the person to each peak (let's call them AC and BC). The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the remaining side.
Let's analyze the situation based on this inequality:
1. Distance from the person to Peak A (AC): Let's assume it is x units.
2. Distance from the person to Peak B (BC): Let's assume it is y units.
3. Distance between the two peaks (AB): Let's call this z units.
According to the triangle inequality, we have:
AC + BC > AB
x + y > z
Now, let's examine the difference of the distances from the person to each peak:
Difference = |AC - BC|
By applying the properties of absolute values, we can rewrite this as:
Difference = |AC - BC| = |x - y|
To prove that the distance between the two peaks (AB) is greater than the difference of the distances from the person to each peak (AC and BC), we need to show that:
AB > Difference = |AC - BC| = |x - y|
However, we cannot make general assumptions about the relative magnitudes of x and y without more specific information. So, it is not always true that the distance between the two peaks will be greater than the difference of the distances from the person to each peak.
In certain cases, it is possible for the distance between the peaks to be smaller than the difference between the distances to each peak. This will occur when the person is closer to one peak than the other.