In this activity, students examine the converse of the Pythagorean Theorem: If the sides of a triangle are of lengths and and , then the triangle is a right triangle. Students do not need to create a rigorous proof of this fact, but the Are You Ready For More? section provides an outline of how the fact can be proven.

Students must make use of the structure of triangles and the relationship between an angle and its opposite side to describe the triangles (MP7).

The purpose of this discussion is to verify the converse of the Pythagorean Theorem as well as to classify triangles as acute, right, or obtuse based on their side lengths.

Invite students to share their observations about triangles and and their reasoning for those observations. Highlight students who classify the triangles as acute, right, or obtuse.

Then, invite students to share their conjectures about triangles based on the inequalities given. Write the conjectures for all to see.

“A triangle has sides of length and so that .

• If , then it is a right triangle.
• If , then it is an acute triangle.
• If , then it is an obtuse triangle.”

Tell students that they will not prove the conjectures, but that the conjectures are true.

If time allows, ask students,

• “How could you verify these statements for equilateral triangles?” (In an equilateral triangle, all the sides are the same length and all angles are 60 degrees, so it is acute. Because all of the sides are the same length and not zero, we can write which matches with the statement that says it is an acute triangle.)
• “A triangle has sides of length 8, 9, and 15. Is this triangle acute, right, or obtuse? Explain your reasoning.” (The triangle is obtuse because .)
• “A triangle has sides of length 10 and 11. How long must the third side be so that the angle between the sides of length 10 and 11 is obtuse?” (Greater than because .)

• “Is it possible for ?”
• “If you use your hands for two of the side lengths and keep your fingertips together, how far apart could your wrists be?”

The purpose of the discussion is for students to understand that the length of any side of a triangle must be less than the sum of the other two sides of the triangle.

Invite students to share their inequality and reasoning.

Ask students,

• “A triangle has sides of length 5 and 9. What is the longest the other side can be? What is the shortest it can be?” (It must be less than 14. It must be greater than 4.)
• “A triangle has sides of length and . What is the longest the other side can be? What is the shortest it can be?” (It must be less than . It must be greater than and , which can be combined as being greater than ).
• “How does the inequality show that the shortest distance between two points is a straight line?” (If another path is taken so that it goes through a point not on the line between the two points, then the sum of the distance from the start to that point plus the distance from that point to the end must be greater than the segment that connects the start and the end.)