An isosceles triangle has two congruent sides called legs and a scalene triangle has no congruent sides.
Creating an isosceles triangle
Let's use the first string to create our isosceles triangle. Note that the Triangle Larger Side Theorem has to hold true for all of the triangle's sides. If we call the legs of the triangle x, then the base side can be expressed as 24-2x.
By the Triangle Larger Side Theorem, we can create the following three inequalities.
& x+ x> 24-2x
& x+( 24-2x)> x
&( 24-2x)+ x> x
Note that the second and third inequality are the same so we only have to solve one of them. By solving both inequalities we get a range of values for x
In our isosceles triangle, x has to be greater than 6 but less than 12. As long as we keep the triangle's legs within this range, we can create an isosceles triangle from our string. For example, the triangle could have two legs that are 7 cm and a base that is 10 cm.
All of the angles are acute so this is an acute isosceles triangle.
Creating a scalene triangle
We will use the second string to create the second triangle. As this is going to be a scalene triangle, no sides can be congruent. For this purpose, we will call the length of one side a, the length of another side we call c, and the third side becomes 24-a-c.
Again, using the Triangle Larger Side Theorem, we can write the following three inequalities.
& a+ c> 24-a-c
& a+( 24-a-c)> c
&( 24-a-c)+ c> a
Let's simplify the first inequality.
