The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
If ΔABC is a right triangle, then a ^{2} + b ^{2} = c ^{2}.
The converse is also true.
If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.
If a ^{2} + b ^{2} = c ^{2}, then ΔABC is a right triangle.
Example Identifying Right Triangles
Decide if each of the following is a right triangle. Justify your reasoning.
a.
b.
c.
d.
Solution
Use the converse of the Pythagorean Theorem to determine whether each triangle is a right triangle.
a.
table with 4 rows and 2 columns , row1 column 1 , , eh squared , plus , b squared , , column 2 modified modified question mark with under bar below with under bar below , , c squared , row2 column 1 , , 6 squared , plus , 8 squared , , column 2 modified modified question mark with under bar below with under bar below , , 10 squared , row3 column 1 , 36 plus 64 , column 2 modified modified question mark with under bar below with under bar below , 100 , row4 column 1 , 100 , column 2 equals 100 u 2 , end table
Yes, the triangle is a right triangle because 6^{2} + 8^{2} = 10^{2}.