Worksheet on triangle inequality property of sides in a triangle
SOLVED PROBLEM 1:
In a triangle ABC, the lengths of the three sides are 7 cms, 12cms and 13cms. Among the following three types of triangles: acute angle triangle , right angled triangle and obtuse angled triangle, which triangle is formed?
To solve this question on finding the type of triangle formed, we must use the triangle inequality property on the sides to determine what type of angles are formed in the triangle.
To learn in detail about the triangle inequality property, click the following link:
Let us recall the property briefly here:
If the square of the length of every side in any triangle is less than the sum of the squares of the lengths of the other two sides in the triangle, then the type of triangle formed is acute angled triangle.
In mathematical inequality form, the above statement (italicized) can be written as follows:
AC 2 < AB 2 + BC 2
Now, among the numbers given in the above question for the lengths of the three sides in the triangle ABC, let us pick 13 as the length of the side AC.
Next, we will square each of the numbers (which represent the lengths of the sides of the triangle ABC) to verify if the above mathematical inequality holds.
AC 2 = 13 2 = 169.
AB 2 = 12 2= 144.
BC 2= 7 2 = 49.
Now, check if the above mathematical inequality conveying the triangle inequality property holds as follows:
Is 13 2 < 12 2 + 7 2 ?
i.e. is 169 < 144 + 49?
And the answer is
So, 13 2 < 12 2 + 7 2 is applicable.
This mathematical inequality in a triangle holds on the side formed opposite the angle which is ACUTE in measure.
Therefore the angle formed opposite the side having the length 13 cms is acute in measure.
(Recall the measure of an acute angle is 0 0 to 90 0 , exclusive).
Also, note that if 13 2 < 12 2 + 7 2, then it is too obvious to verify
Whether 12 2 < 13 2 + 7 2 and 12 2 < 13 2 + 7 2
Therefore, to find what angle is formed opposite any side in a triangle in which the lengths of the three sides are given, it is enough to check only if:
The Square of the biggest number is less than or equal to or greater than the sum of the squares of the other two numbers.(as for instance in the above example, it is just sufficient to check if 13 2 < 12 2 + 7 2).
Hence all the three angles in the given triangle ABC in which the lengths of the sides are 7, 12 and 13 are all acute angles and therefore the triangle can be defined as acute angled triangle in nature.
SOLVED PROBLEM NO. 2:
In a triangle PQR, the lengths of the sides are the following numbers:
p, p + 5 and 20 – p.
then which of the following numbers p cannot be?
6, 8, 10, 12 and 15.
To find which of the numbers in the answer options in the above question cannot be a possible value of the length of a side, we must apply the popular triangle inequality theorem on the length of any side in a triangle which runs as follows:
The length of any side in any triangle must be greater than the difference of the lengths of the other two sides in the triangle and also be less than the sum of the lengths of the other two sides in the same triangle.
Simply put in a mathematical inequality form, the above statement of inequality on the lengths of the sides in any triangle can be written as below:
a – b < c < a + b
in the above mathematical inequality laying conditions on the lengths of the three sides in any triangle, a, b and c represent the lengths of the three sides in the triangle.
The inequality above arises to enable a proper formation of a triangle given the lengths of the three sides in a triangle.
Because if the length of the side in a triangle is too short, then a triangle is not properly formed and again, when the length of the side is extremely long, also then a triangle will not be properly formed.
The following figure illustrates what is stated in italics above:
The above inequality is popularly called the triangle inequality theorem applicable on the lengths of the sides in any triangle.
Every side in every side must obey the above theorem.
Even the greatest side, the hypotenuse in a right angled triangle must obey the above triangle inequality property and even the greatest side opposite the obtuse angle must fit in with the property.
For a detailed discussion of the above property, do click the following link:
Now that you have read in detail about the triangle inequality property, let us proceed to apply it in solving the above question to find which of the numbers in the answer options cannot be a possible value of the length of the side in a triangle in which the lengths of the three sides in the triangle are given to be:
p, p + 5 and 20 – p
Now, from among the three numbers
p, p + 5 and 20 – p,
one can choose any number for a and the other for b and the third for c.
Still, we will choose
p + 5 for a and the other two numbers
20 – p and p
for the other two lengths of the sides namely b and c in the triangle.
Now let us apply the given numbers for the lengths of the sides of the triangle:
a, b and c as follows:
(Note: In the full triangle inequality theorem for determining the lengths of the sides which runs as a – b < c < a + b, we will pick only the right side part which is c < a + b as follows
p + 5 < 20 – p + p
p + 5 < 20 – p + p
p + 5 < 20
p < 20 – 5
p < 15
thus we find that the length of one side of a triangle cannot be as long as 15 if a triangle has to be indeed properly formed having the three numbers:
p, p + 5 and 20 – p
as the lengths of the sides of the triangle.
(For, if you put 15 in p, then the three number p, p + 5 and 20 – p would become respectively 15, 20 and 5. So, what we have is the length of a side in the triangle namely 20 is equal to the sum of the other two numbers : 5 and 15, instead of being less than their sum as per the triangle inequality theorem : a < b + c)
Thus, the triangle inequality theorem of sides can be used to determine what upper limit and lower limit the lengths of the sides in a triangle should have in order to enable a proper formation of a triangle.