Question

5. Given the following triangle side lengths, identify the triangle as acute, right or obtuse. Show your work.

Answer 1

Refer to the figure shown below.
We shall review each of the three given measurements and decide what type of triangle we have.

Measurement a.
a=3, b=4, c=5.
For a right triangle, c² = a² + b² (Pythagorean theorem)
a² + b² = 3² + 4² = 9 + 16 = 25
c² = 5² = 25
Answer:
This is a right triangle, because c² = a² + b².

Measurement b.
a=5, b=6, c=7.
For an acute triangle, c² < a² + b².
a² + b² = 5² + 6² = 25 + 36 = 61
c² = 7² = 49
Answer:
This is an acute triangle, because c² < a² + b².

Measurement c.
a=8, b=9, c=12.
For an obtuse triangle, c² > a² + b².
a² + b² = 8² + 9² = 64 + 81 = 145
c² = 12² = 144
Answer:
This is an acute triangle because c² < a² + b².

Answer 2

Answer:

Let "a", "b" and "с"  be sides of the triangle ("с" is the longest side).

The triangle will be:

right if       a² + b² = c²

аcute if     a² + b² > c²    

obtuse if   a² + b² < c²    

a.

a=3, b=4 and c=5

a² + b² = 3² + 4² = 9 + 16 = 25   and   c² = 5² = 25

25 = 25   ⇒  right triangle.

b.

a=5, b=6 and c=7

a² + b² = 5² + 6² = 25 + 36 = 61   and   c² = 7² = 49

61 > 49   ⇒  аcute triangle.

c.

a=8, b=9 and c=12

a² + b² = 8² + 9² = 64 + 81 = 145   and   c² = 12² = 144

145 > 144   ⇒  аcute triangle.