Problem 4 Identifying a Right Triangle

A triangle has side lengths 85, 84, and 13. Is the triangle a right triangle? Explain.

a 2 + b 2 = ? c 2 Pythagorean Theorem 13 2 + 84 2 = ? 85 2 Substitute  13  for  a ,   84  for  b ,  and  85  for  c . 169 + 7056 = ? 7225 Simplify . 7225 = 7225 ✔   table with 4 rows and 3 columns , row1 column 1 , eh squared , plus , b squared , column 2 modified equals with question mark above , c squared , column 3 cap pythagoreancap theorem , row2 column 1 , 13 squared , plus , 84 squared , column 2 modified equals with question mark above , 85 squared , column 3 cap substitute . 13 , for , eh comma 84 , for , b comma , and , 85 , for , c . , row3 column 1 , 169 plus , 7056 , column 2 modified equals with question mark above , 7225 , column 3 cap simplify , . , row4 column 1 , 7225 , column 2 equals , 7225 , u 2 , column 3 , end table

Yes, the triangle is a right triangle because 13 2 + 84 2 = 85 2 . 13 squared , plus , 84 squared , equals , 85 squared , .

✓ Got It?

4. 1. A triangle has side lengths 16, 48, and 50. Is the triangle a right triangle? Explain.
   2. Reasoning Once you know which length represents the hypotenuse, does it matter which length you substitute for a and which length you substitute for b? Explain.

Take Note Theorem 8-3

Theorem

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.

If …

c 2 > a 2 + b 2 c squared , greater than , eh squared , plus , b squared

Then …

Δ A B C cap delta eh b c is obtuse

You will prove Theorem 8-3 in Exercise 53.

Theorem 8-4

Theorem

If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.

If …

c 2 < a 2 + b 2 c squared , less than , eh squared , plus , b squared

Then …

Δ A B C cap delta eh b c is acute

You will prove Theorem 8-4 in Exercise 54.