Two altitudes of a triangle have lengths 12 and 14. What is the longest possible length of the third altitude, if it is a positive integer?
I tried using the triangle inequality but to no avail...how would I do this?
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To find the longest possible length of the third altitude, we can use the concept of the triangle inequality. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's assume that the two altitudes of the triangle with lengths 12 and 14 are bases of the triangle. Without loss of generality, let's consider the base with length 12.
Since it is given that the third altitude is a positive integer, let's consider the smallest possible value for the third altitude. The smallest positive integer greater than 0 is 1. Therefore, let's consider a triangle with base lengths of 12 and 14, and assume that the length of the third altitude is 1.
Using the triangle inequality, we have:
12 + 14 > 1
Simplifying the inequality:
26 > 1
Since this inequality is true, we know that a triangle with base lengths of 12 and 14, and a third altitude with a length of 1, is possible.
Now, let's consider whether a triangle with a base length of 12 and a third altitude length of 2 is possible or not.
Using the triangle inequality, we have:
12 + 2 > 14
Simplifying the inequality:
14 > 14
This inequality is not true, which means a triangle with a base length of 12 and a third altitude length of 2 is not possible.
Similarly, we can continue this process of checking the triangle inequality for increasing values of the third altitude until we find the largest possible length for the third altitude that satisfies the triangle inequality.
By following this process, you should find the longest possible length of the third altitude.