Construct a triangle with constraints | Geometry | 7th grade | Khan Academy

TL;DR

The video explains why it is impossible to construct a triangle with side lengths of 2, 2, and 5, and explores the conditions for constructing non-degenerate triangles.

Transcript

If someone walks up to you on the street and says, all right, I have a challenge for you. I want to construct a triangle that has sides of length 2. So sides of length-- let me write this a little bit neater. Sides of length 2, 2, and 5. Can you do this? Well, let's try to do it. And we'll start with the longest side, the side of length 5. So the s... Read More

Key Insights

• 🪘 The longest side of a triangle cannot be longer than the sum of the other two sides.
• 🫥 Degenerate triangles have no area and resemble line segments.
• 🪘 A non-degenerate triangle can be constructed if the sum of the other two sides is longer than the longest side.

Questions & Answers

Q: Can a triangle be constructed with side lengths of 2, 2, and 5?

No, because the longest side (5) is longer than the sum of the other two sides (2 + 2 = 4), violating the triangle inequality theorem.

Q: What is a degenerate triangle?

A degenerate triangle is a triangle that has no area and looks like a line segment. It occurs when the longest side is exactly equal to the sum of the other two sides.

Q: How can a non-degenerate triangle be constructed with side lengths of 3, 3, and 5?

By rotating the sides of length 3 all the way inwards, the points will touch, creating a non-degenerate triangle. The angle between the two sides of length 3 becomes 0 degrees.

Q: Is the triangle with sides of length 3, 3, and 5 unique?

Yes, in this case, the triangle is unique. The points where the sides touch are fixed, and no other configuration is possible due to the given side lengths.

Summary & Key Takeaways

• The video demonstrates how to construct a triangle and starts by attempting to construct a triangle with side lengths of 2, 2, and 5.

• It explains that the longest side of a triangle cannot be longer than the sum of the other two sides.

• The video also introduces the concept of degenerate triangles, where the triangle becomes a line segment, and discusses the conditions for constructing non-degenerate triangles.