What is the reflexive property of congruence?
The triangle inequality theorem states that for any given triangle, the sum of any two sides is always greater than the third side. It describes the relationship between the sides of the triangle and is applicable for any type of triangle be it equilateral, isosceles or scalene.
Consider a ∆ABC as shown where a, b, and c are the length of the sides.
According to the theorem ABC is a triangle only if,
a+b>c,
b+c>a,
c+a>b.
How can we use the triangle inequality theorem?
It is a very simple theorem to use. Whenever 3 lengths are given, we need to just verify if the sum of any 2 is always greater than the third side.
Suppose, we need to check 11,13 and 15 can form a triangle
11+13 >15
13+15>11
15+11>13
So, there exists a scalene triangle with the sides 11,13 and 15 units.
If side lengths are given to be 4, 4, and 8 units, these form 3 collinear points since,
4+4 = 8 (the triangle inequality theorem fails for one set as the sum should be greater than third)
So it's not possible to have a triangle with these dimensions. However, 5,5 and 8 are possible.
Note that in a right triangle with sides 3, 4, and 5 or any equilateral triangle too, this constraint will be met.
Sample Math Problems
Question 1: Can line segments of length 3 cm, 7 cm and 6 cm be used to form a triangle? Check if 4.1 cm, 6.9 cm and 2.81 cm measures, forms a triangle?
Answer:
3+7 > 6
7 +6 >3
6+3 >7
Yes they can form a triangle.
For the second instance,
4.1 + 6.9 = 11 >2.81
6.9 + 2.81 = 9.71 > 4.1
4.1 + 2.81 = 6.91 > 6.9
Yes they form a triangle.
Question 2: If 2 sides of a triangle are 5 and 8 units, find all possible lengths of the third side.
Answer:
Let the third side be x.
x< (5 +8) since 5+8 > x for triangle to exist.
x <13.
But 5+x>8 which means x>8-5 which is x>3
Also, 8+x>5 which means x>5-8 which is x>-3
x>3 and x>-3 implies that x>3
So, x can take any value from 3 to 13 units.
Question 3: Do the points (1,3) (3,5) and (5,7) form a triangle?
Answer:
We need to first find the length of the sides using the distance formula and then apply the inequality theorem.
AB = √(1-3)2 +(3-5)2 = √4+4 = 2.83
BC = √(3-5)2 +(5-7)2 = √4+4 = 2.83
AC = √(1-5)2 + (3-7)2 = √16+16 = 5.66
AB+BC = AC which does not satisfy the inequality theorem. All points are collinear.
Collinear points don't form a triangle.
Question 4: Is it possible to have a hypotenuse of an isosceles triangle double the length of the other 2 sides?
Answer:
No, because shorter sides of the triangles are equal in isosceles and if each is x,
Then x +x =2x and double the length of shorter sides is also 2x
But sum of any two sides > third side(hypotenuse).
So it is not possible to have such a triangle.
