Extension to the Pythagorean Theorem
Variations of Theorem 66 can be used to classify a triangle as right, obtuse, or acute.
Theorem 67: If a, b, and c represent the lengths of the sides of a triangle, and c is the longest length, then the triangle is obtuse if c2 > a2 + b2, and the triangle is acute if c2 < a2 + b2.
Figures 1 (a) through (c) show these different triangle situations and the sentences comparing their sides. In each case, c represents the longest side in the triangle.
![](http://cliffsnotescms-v1.stage.techspa.com/assets/18365.jpg)
Figure 1 The relationship of the square of the longest side to the sum of the squares of the other two sides of a right triangle, an obtuse triangle, and an acute triangle.
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/18352.jpg)
Example 1: Determine whether the following sets of three values could be the lengths of the sides of a triangle. If the values can be the sides of a triangle, then classify the triangle. (a) 16‐30‐34, (b) 5‐5‐8, (c) 5‐8‐15, (d) 4‐4‐5, (e) 9‐12‐16, (f) ![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/18375.jpg)
(Recall the Triangle Inequality Theorem, Theorem 38, which states that the longest side in any triangle must be less than the sum of the two shorter sides.)
a. ![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/18376.jpg)
This is a right triangle. Because its sides are of different lengths, it is also a scalene triangle.
b. ![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/18377.jpg)
This is an obtuse triangle. Because two of its sides are of equal measure, it is also an isosceles triangle.
c. ![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/18378.jpg)
d. ![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/18379.jpg)
This is an acute triangle. Because two of its sides are of equal measure, it is also an isosceles triangle.
e. ![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/18380.jpg)
This is an obtuse triangle. Because all sides are of different lengths, it is also a scalene triangle.
f. ![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/18381.jpg)
This is a right triangle. Because two of its sides are of equal measure, it is also an isosceles triangle.