More On The Triangle Rule

Yesterday we discussed the triangle rule, and I provided a few problems from standardized tests where the triangle rule was extremely useful.  and it was not hard to see how this simple rule should be applied.

Recall that the triangle rule states that that the length of the third side of a triangle is between the sum and difference of the lengths of the other two sides.

This can be written symbolically as “difference < x < sum” where x is the third side of the triangle.

Today let us look at another type of problem that the triangle rule should be used to solve.

Problems involving distances between three points can often be solved using the triangle rule. After all, when you plot three points and look at the distances between each pair of points, you are looking at the lengths of the sides of a triangle.

Try plotting three points on a piece of paper and you you will immediately see what I mean.

There is a small catch however – sometimes when you plot three points, the points can be collinear: that is they all lie on the same line.

This means that the symbol “<” in the triangle rule should be replaced by the symbol “≤.”

In other words, this time we write “difference ≤ x ≤ sum.”

Let’s look at an example that could appear on a standardized math test.

Points Q, R and S lie in a plane. If the distance between Q and R is 18 and the distance between R and S is 11, which of the following could be the distance between Q and S?

A) I only
B) II only
C) III only
D) I and III
E) I, II and III

Solution: Okay, to solve this problem, of course we are going to use the triangle rule… In this case, if Q, R and S form a triangle, then the length of QS is between 18 – 11 = 7 and also 18 + 11 = 29. The extreme cases 7 and 29 form straight lines. In this problem that is fine, so the distance between Q and S is between 7 and 29, inclusive. Thus, the answer is choice (E).

You can see how quickly this problem can be solved when the triangle rule is used.

If you want more practice, then please check out the  Get 800 collection of test prep books. These have more examples of problems that need to use this rule for efficient and correct answering.

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