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The Range of the Third Side

Part of the Triangles section of Coddy's Geometry journey. Lesson 3 of 60.

Take two sides of 9 and 4. The third side cannot be 13, because $9+4 = 13$ . The three sides would lie flat. It cannot be 5 either, because $4+5 = 9$ . Again they would lie flat. Any length between 5 and 13 works.

$9-4 < x < 9+4$

So $5 < x < 13$ . The third side is longer than the difference of the two sides and shorter than their sum. The difference is the bigger side minus the smaller one. For any two sides a and b, the third side c follows this rule. Here $\left|a-b\right|$ means the difference:

$\left|a-b\right| < c < a+b$

Challenge

Two sides of a triangle are 11 and 4. The third side is x. Write all the lengths x can have as one line, like 5 < x < 13. Then press Check.

Start the lesson to read it in the interactive player and track your progress.

All lessons in Triangles

1The Triangle Inequality

Can Three Sticks Meet?Exactly ReachingThe Range of the Third SideRecap: The Third Side

4Area of a Triangle

Half the Rectangle Height Is Not a Side Right Triangle Area Finding the Height Any Side Can Be the Base Recap: Area

7Is It Right?

The Converse Testing a Triangle Acute or Obtuse?Sorting by the Test Recap: Right or Not

10Scale in the World

Shadows Across the River Nested Triangles A Side with an Expression Recap: Scale in the World

2Sides and Names

Sorting by Sides The Longest Side Ticks into Equations Recap: Sorting Triangles

5The Pythagorean Theorem

Squares on the Sides Finding the Hypotenuse Pythagorean Triples When It Is Not Whole 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

11Two Tools at Once

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