Tell if the measures 18, 20 and 27 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse or right. Please explain.
In a triangle the sum of any two sides of the triangles is greater than the third side. A triangle can be formed by any set of three lines that satisfy this condition.
As we see below this condition is satisfied for the set of given lines that have length of 18, 20 and 27.
18 + 20 > 27
20 + 27 > 18
18 + 27 > 20
In an triangle the biggest angle is formed by the two smallest sides. We can find if this angle is equal to, less than, or more than right angle as follows.
Let the two smaller sides be equal to a and be and the biggest side be c. Then the angle formed by a and b is:
- Right angle if a^2 + b^2 = c^2
- Obtuse angle if a^2 + b^2 < c^2
- Acute angle if a^2 + b^2 > c^2
For the given three lines:
a^2 + b^2 = 18^2 + 20^2 = 724
c^2 = 27^2 = 729
Thus we see that a^2 + b^2 < c^2. Therefore the angle is an obtuse angle, and the triangle formed by the lines is an obtuse triangle.
The condition for the given 3 lengths form a triangle is that the greatest of the 3 sides should be less than the other two sides. Or if a is the greatest of the 3 sides and the other two sides are b and c, then the greatest side a is less than the sum of the other two sides:
a < b+c.
In the given case a = 27, b= 18 and c= 20.
b+c = 20+18 = 38. a= 27.
So 27 < 38. So the given 3 lengths can form a triangle.
If a = b + c, we get a triangle of zero area.
If a > b > c and a < b + c , the formation of triangle is not possible.