Hard math Question (with triangles!)

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1

<p>Points A,B, and C lie in a plane. If the distance between A and B is 7 and the distance between B and C is 4 , which of the following could be the distance between A and C. </p>

<p>I. 3
II. 10
III. 11</p>

<p>(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
Here's how I did the problem. The book (32 most effective math strategies) states "the third side of a triangle is between the sum and difference of the other two sides" .</p>

<p>so
7+4 =11
7-4=3</p>

<p>According to the theory , third side should be between 3-11. I also learned in school that the two smaller legs of the triangle, added together, must be larger than the largest leg. That narrowed my choice to 10 and picked choice (B)</p>

<p>When I looked at the answer explanation, it said the answer was choice (E). It explained that "the extreme cases 3 and 11 form straight line. In this problem that is fine , so the distance between A and C is between 3 and 11 INCLUSIVE" </p>

<p>Can someone please how you know 3 and 11 forms a straight line, why that is fine in this problem, and how to identify exceptions to the rule such as these? </p>

<p>Thanks in advance and Good Luck tomorrow everyone!</p>

2

<p>This is easy to see with a simple picture.</p>

<p>Draw a line segmant AB, and label it with a 7. Now extend the segment from B to C so that you form a line segment that starts with A, passes through B and ends at C. Label BC with a 4. Now note that the length of AC is 7 + 4 = 11. </p>

<p>This doesn’t form a triangle (it forms a line segment), but as far as distances go it makes perfect sense.</p>

<p>The other case is slightly trickier. Draw segment AB as in the last case, and again label it with a 7. But this time draw BC by tracing backwards along segment AB, and label BC with a 4. The picture should be a segment that begins with A, passes through C and ends at B. The whole segment AB has length 7, and the segment CB has length 4. Thus AC has length 7 - 4 = 3.</p>

<p>Remark: A line is a “degenerate” triangle. In other words if you collapse a triangle it becomes a line.</p>

<p>Bottom line: If the problem is specifically about triangles, then the inequalities should be strict. If the problem is about distances, then the extreme values work as well.</p>

<p>If you need further clarifying let me know, and best of luck tomorrow!</p>