Video Transcript
Consider triangle 𝐴𝐵𝐶 with 𝐴𝐵 equals nine, 𝐵𝐶 equals 10, and 𝐴𝐶 equals 11. What kind of triangle is this in terms of its angles?
The first thing we might want to think about are the types of triangles in terms of their angles. We have a right-angled triangle where one of the angles measures 90 degrees, an obtuse triangle where one angle measures greater than 90 degrees, and an acute triangle where all three angles must be less than 90 degrees.
We also know one formula that helps us relate side lengths to types of triangles by their angle measure. And that’s the Pythagorean theorem that’s used for right-angled triangles. In a right-angled triangle, the side length opposite the right angle is the longest side length. And we know that that side length squared, 𝑐 squared, is equal to 𝑎 squared plus 𝑏 squared.
We can relate the Pythagorean theorem to obtuse triangles or acute triangles. In an obtuse triangle, the square of the longest side will be greater than adding the two squares of the shorter sides. And in a similar way, in an acute triangle, the square of the longest side will be less than the sum of the squares of the two smaller sides. We’ve been given three side lengths about the triangle 𝐴𝐵𝐶. To find out what type of triangle this is, we want to square all of the sides then add the two smaller sides together.
Is 121 equal to 81 plus 100? 121 is not equal to 181. 121 is less than 181. And since the square of the longest side is smaller than the squares of the other two sides added together, we know that this is going to be an acute triangle. So, we can say that triangle 𝐴𝐵𝐶 is an acute-angled triangle.
