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The Range of the Third Side

Part of the Triangles section of Coddy's Geometry journey. Lesson 11 of 71.

Now put the two limits together. With sides 6 and 10, the third side x must be more than their difference and less than their sum. You can write both conditions as one chain:

10 - 6 < x < 10 + 6

4 < x < 16

Every value of x strictly between 4 and 16 makes a triangle with the 6 and the 10. No other value does.

Solve the chain as one line, as you did in the inequalities section. Work out the arithmetic in each outer part, and the middle part stays x. The answer is a range: the third side is longer than the difference and shorter than the sum of the two known sides. When a figure shows two known sides and labels the third side x, it is asking for this range.

$A triangle with sides labelled 10, 6 and x$

With sides 10 and 6, the third side x lies between 4 and 16

Challenge

A triangle has sides 12, 5 and x.

$A triangle with sides labelled 12, 5 and x$

Find the range of x.

Try it yourself

12 - 5 < x < 12 + 5

Solve for x

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Triangles

2The Triangle Inequality

Can Three Sticks Meet?Exactly Reaching How Short Can It Be?How Long Can It Be?The Range of the Third SideThe Formula Whole-Number Sides Recap: The Third Side

5The Hypotenuse

Squares on the Sides Finding the Hypotenuse Pythagorean Triples When It Is Not Whole Simplifying the Root Recap: The Hypotenuse

8Special Right Triangles

The 45-45-90 Triangle Legs from the Hypotenuse The 30-60-90 Triangle All Three Sides Which Triangle Is It?Recap: Special Triangles

11Mixing Area and Pythagoras

Height of an Isosceles Area from the Height The Equilateral Triangle Right Triangle Perimeter Recap: Two Tools