Understanding Acute Triangles

Acute triangles have a special feature: all three of their inside angles are less than 90 degrees. This means that, like all triangles, the total of the angles adds up to 180 degrees, but none of the angles can be a right angle (which is exactly 90 degrees).

Characteristics of Acute Triangles:

1. Angle Measures: All inside angles are acute, meaning each angle is less than 90 degrees.

2. Side Lengths: Acute triangles can have any combination of side lengths. However, they usually don’t have a longest side that is equal to or longer than the average of the other two sides.

3. Types:

   • Acute triangles can be equilateral (all sides and angles are equal, with each angle measuring 60 degrees).
   • They can also be isosceles (two sides are equal, and all angles are less than 90 degrees).

4. Circumcircle: An acute triangle can fit inside a circle that touches all its corners, and the circle’s center is located inside the triangle.

Using the Pythagorean Theorem:

The Pythagorean Theorem is a rule that helps us with right triangles. It says that if you take the lengths of the two shorter sides (called legs) of a right triangle, square them, and add those numbers together, you’ll get the square of the longest side (called the hypotenuse). Here’s how it looks:

$$a^2 + b^2 = c^2$$

In this formula, $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.

How to Tell Acute Triangles Apart from Right and Obtuse Triangles:

1. For Acute Triangles:

   • If you have sides with lengths $a$, $b$, and $c$ (with $c$ being the longest), then the triangle is acute if:
   $$a^2 + b^2 > c^2$$

2. For Right Triangles:

   • The triangle is called right if:
   $$a^2 + b^2 = c^2$$

3. For Obtuse Triangles:

   • It’s obtuse if:
   $$a^2 + b^2 < c^2$$

In short, the way the side lengths relate according to the Pythagorean Theorem helps us figure out the type of triangle we have. This clearly sets apart the characteristics of acute triangles from the others.