Question

# A classmate drew an acute triangle with sides 9 in. and 12 in. What is the greatest possible whole number that can be the length of the longest side o

Right triangles and trigonometry
A classmate drew an acute triangle with sides 9 in. and 12 in. What is the greatest possible whole number that can be the length of the longest side of the triangle in inches? Provide evidence.

2021-06-04

By the Pythagorean Inequality Theorem, a triangle is acute if $$a2+b2>c2a$$ where c is the longest side. So, we can write
$$\displaystyle{9}^{{2}}+{12}^{{2}}{>}{c}^{{2}}$$
$$\displaystyle{81}+{144}{>}{c}^{{2}}$$
$$\displaystyle{225}{>}{c}^{{2}}$$
$$\displaystyle{15}{>}{c}$$
This means that the longest side must be less than 15 inches. Hence, the e greatest possible whole number that can be the length of the longest side of the triangle is 14 inches.