hi guys ive be struggling on this problem for a couple of days, so please help if you can
Problem: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?
Thanks in advance and merry christmas: happy new year tomorrow!
This is a question from the Intermediate Algebra homework. People shouldn't be asking for the answer on different websites. Thanks!
-Art of Problem Solving
To solve this problem, we need to use a property of altitudes in a triangle. The property states that for a triangle, the product of any two altitudes is equal to twice the area of the triangle.
Let's say the lengths of the two given altitudes are a and b, with a = 12 and b = 14. Let h be the length of the third altitude. According to the property mentioned above, we have:
a * b = 2 * Area of the triangle
12 * 14 = 2 * Area of the triangle
168 = 2 * Area of the triangle
To find the longest possible length of the third altitude, we need to maximize the area of the triangle. The area of a triangle can be calculated using the formula:
Area of the triangle = (base * height) / 2
In this case, since the lengths of the altitudes represent the heights of the triangle, we can calculate the area as follows:
Area of the triangle = (12 * 14) / 2
Area of the triangle = 168 / 2
Area of the triangle = 84
Now, let's determine the base of the triangle. Since the third altitude is the longest possible length, it forms the base of the triangle. We'll label the base as b1.
Area of the triangle = (b1 * h) / 2
84 = (b1 * h) / 2
To find the longest possible length of the third altitude (or h), we need to consider the factors of 84. The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84. However, we are looking for a positive integer, so we can narrow down the possible values for h.
Since the base (b1) and height (h) are integers, we need to find a factor pair of 84 where one factor is greater than 14 (since the two given altitudes are 12 and 14). From the list, we find that the factor pair (21, 4) satisfies this condition.
Therefore, the longest possible length for the third altitude is 21 units.
Hope this helps! If you have any further questions, feel free to ask. Merry Christmas and Happy New Year to you too!