Determine if the following lengths make an acute, right or obtuse triangle. 6. 4, 20, 21.

To determine the type of triangle formed by the sides 4, 20, and 21, you can use the triangle inequality theorem, which states the sum of the lengths of any two sides must be greater than the length of the third side. Here, 4 + 20 = 24, which is greater than 21. Then, 4 + 21 = 25, greater than 20. Finally, 20 + 21 = 41, greater than 4. So these lengths can indeed form a triangle! Next, apply the converse of the Pythagorean theorem. For an acute triangle, \(a^2 + b^2 > c^2\); for a right triangle, \(a^2 + b^2 = c^2\); and for an obtuse triangle, \(a^2 + b^2 < c^2\). Using 4, 20, and 21 with 21 as the longest side: 4² + 20² = 16 + 400 = 416 and 21² = 441. Since 416 < 441, this means the triangle is obtuse. An obtuse triangle has one angle greater than 90 degrees, making it perfect for a dramatic scene or a cool visual effect in your next art project—just think of those bold angles! Want a fun fact? The oldest known triangle is believed to be over 4,000 years old, found in ancient Egypt—proof that triangles have been impressing and baffling us for ages!