Emily and her friend Yesenia have an assortment of building logs. They want to make triangles using the building logs. Emily randomly selects three building logs from the box and wants to make a triangle using them.

Emily has selected the following building logs to investigate the question that Yesenia asked her.

Use the interactive sketch below to investigate which sets of building logs Emily could use to make a triangle. Use the side lengths that are given to see if you can construct a triangle with those side lengths. Make a T-chart of combinations of side lengths that make a triangle and those that do not make a triangle. Use your T-chart to answer the questions that follow.

• 4.5 inches, 4.5 inches, 10 inches
• 4.5 inches, 5 inches, 6.5 inches
• 6.5 inches, 8.5 inches, 11 inches
• 3.5 inches, 3.5 inches, 8.5 inches
• 3.5 inches, 5 inches, 10 inches
• 4.5 inches, 5 inches, 8.5 inches

Interactive popup. Assistance may be required.

Need additional directions?

Interactive popup. Assistance may be required.

See a Sample T-Chart

Use what you noticed in the interactive to answer the following questions.

• For the side lengths that create a triangle, how does the sum of the lengths of the two shorter sides compare to the length of the third side?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The sum of the lengths of the two shorter sides is greater than the length of the third side.

• For the side lengths that do not create a triangle, how does the sum of the lengths of the two shorter sides compare to the length of the third side?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The sum of the lengths of the two shorter sides is less than or equal to the length of the third side.

Pause and Reflect

If you know the lengths of three segments, how can you determine whether or not the segments will create a triangle?

Interactive popup. Assistance may be required.

Check Your Answer

Add the lengths of the two shorter segments. If the sum is greater than the length of the third side, then the segments will create a triangle.

If you know the lengths of two sides of a triangle, how can you determine the possible lengths for the third side of the triangle?

Interactive popup. Assistance may be required.

Check Your Answer

The length of the third side must be less than the sum of the lengths of the two known sides. The length of the third side must also be greater than the difference between the lengths of the two known sides.

The relationship between the lengths of three sides of a triangle is very important. Architects, engineers, and artists use this relationship to make sure that triangular structures are, indeed, full triangles.

Practice

1. Which of the followings combinations of side lengths will create a triangle? Use the interactive presented in this section of the lesson to help you if necessary.
   1. 3 cm, 7 cm, 15 cm
   2. 5.5 cm, 9 cm, 12 cm
   3. 41 over 4 1 4 in., 37 over 8 7 8 in., 6 in.
   Interactive popup. Assistance may be required.

   Need a hint?

   In a triangle, the sum of any two side lengths must be greater than the length of the third side.
   Interactive popup. Assistance may be required.

   Check Your Answer

   II and III only
2. Madison has three pieces of lumber that measure 5.6 feet, 4 feet, and 3.4 feet. Without cutting the pieces of lumber, will Madison be able to create a triangular flowerbed in her front yard? Explain your answer.

   Interactive popup. Assistance may be required.

   Need a hint?

   In a triangle, the sum of any two side lengths must be greater than the length of the third side.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Yes. 3.4 feet + 4 feet = 7.4 feet, which is greater than 5.6 feet.
3. The map below shows the approximate driving distances for the most direct path between three towns and cities in south Texas.
   

   Jacob and his cousin Ethan each left Three Rivers for Corpus Christi. Jacob drove directly to Corpus Christi, and Ethan drove to Alice first to pick up their grandmother. Who drove a longer distance, and by how many miles?

   Interactive popup. Assistance may be required.

   Need a hint?

   In any triangle, the length of the third side must be less than the sum of the lengths of the two remaining sides.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Ethan drove a longer distance by 22.8 miles.