Compare the square of the largest side length to the sum of the squares of the other two side lengths.
Is a triangle? Yes Classification: Right Explanation: See solution.
Practice makes perfect
We want to determine whether the given side lengths can be the measures of a triangle.
To do so, we will use the following theorem.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Let's check if that is the case.
4.5+20 ? > 20.5
20.5+4.5 ? > 20
20+20.5 ? > 4.5
24.5 > 20.5
25 > 20
40.5 > 4.5
Therefore, these sides lengths can indeed form a triangle. Now, we want to determine whether the triangle formed by the given side lengths is acute, right, or obtuse. To do so, we will compare the square of the largest side length to the sum of the squares of the other two side lengths. Let a, b, and c be the lengths of the sides, with c being the longest.
Condition
Type of Triangle
a^2+b^2 < c^2
Obtuse triangle
a^2+b^2 = c^2
Right triangle
a^2+b^2 > c^2
Acute triangle
Let's now consider the given side lengths 4.5, 20, and 20.5. Since 20.5 is the greatest of the numbers, we will let c be 20.5. We will also arbitrarily let a be 4.5 and b be 20.
4.5^2+20^2 ? 20.5^2
Let's simplify the above statement to determine whether the left-hand side is less than, equal to, or greater than the right-hand side.
