ACT Math: The Triangle Inequality

We all know that a triangle is made up of three sides. But do the lengths of those sides matter? Or can any three side lengths make up a triangle?

You might remember your Geometry teacher talking about the Triangle Inequality Theorem. It tells us that some triangles just aren’t possible. Imagine if I gave you three sticks with lengths of 10 inches, 2 inches and 5 inches and I told you to make a triangle. Try as hard as you’d like; you won’t be able to do it. You’ll end up instead with something that looks like this:

Triangle Inequality

So how do you determine if you have three sides that can form a triangle? The three sides must have this property:

Sum of Two Smaller Sides > Largest Side

Let’s see how this works in a problem:

Which of the following could represent the three sides of a triangle?

I.  3 cm, 5 cm, 7 cm
II.  8 inches, 12 inches, 23 inches
III.  2 feet, 2.6 feet, 3.5 feet

A.  I only
B.  II only
C.  III only
D.  I and II
E.  I and III

Let’s test each one using the inequality above

I.  3 + 5 > 7   True!
II.  8 + 12 > 23 False! This can’t be a triangle
III. 2 + 2.6 > 3.5  True!
 

Only I and III could represent the sides of triangle, so the correct answer is E.

Not bad! But how about the case where we know two of the sides (Let’s call them a and b) and we want to know the possible lengths of the third side (Let’s call it c). For this, we need another inequality:

a – b < c < a + b

So, for instance, what if I tell you that two sides of a triangle are 8 inches and 11 inches and I ask you to tell me the possible lengths of the third side.

11 – 8 < c < 11 + 8
3 < c < 19

The third side must have a length that is between 3 inches and 19 inches (but not 3 inches or 19 inches).

Here’s a problem for you.

Two sides of a triangle are 7 inches and 10 inches. Which of the following is a possible length for the third side of the triangle?

I.  3 inches
II.  15 inches
III. 19 inches

A.  I only
B.  II only
C.  III only
D.  I and II
E.  II and III

Let’s use our inequality to find the possible lengths for that third side.

a – b < c < a + b
10 – 7 < c < 10 + 7
3 < c < 17

The third side must be between 3 inches and 17 inches (but can’t equal either 3 inches or 17 inches). Of the three choices, the only one that works is 15 inches. The correct answer is Choice B.

If you have questions about the Triangle Inequality or anything else having to do with the ACT, send us an email at info@cardinalec.com

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